Dynamical Modelling and Controllability Analysis of an Underactuated 2-Dimensional TORA System on a Slope

Abstract

The 2-Dimensional Translational Oscillators with Rotating Actuator (2DTORA) is a novel underactuated system which has one actuated rotor and two unactuated translational carts. This paper focuses on dynamical modelling and simulation analysis of the underactuated 2DTORA on a slope. Based on Lagrange equations, the dynamics of the 2DTORA is achieved by selecting a transverse position of a cart, a travelling position of a cart, and the rotor angle as the general coordinates and torque acting on the rotor as the general force. When the slope angle is set to zero, the dynamics of 2DTORA on a slope is reduced to that of 2DTORA on the horizontal plane. Moreover, by eliminating one degree of translational cart motion, the dynamics of 2DTORA is reduced to that of TORA which is a benchmark of underactuated systems. In addition, the equilibrium and controllability of the 2DTORA system on a slop are discussed. Finally, numerical simulations are performed to verify the feasibility of the developed dynamic models.

Keywords

Underactuated System 2DTORA Dynamic Modelling Controllability TORA

1. Introduction

Underactuated systems are a class of mechanical systems whose control inputs are less than the number of configuration variables. Because underactuated systems need less control inputs than the fully-actuated systems, they have advantages in saving energy, reducing the cost, and enhancing the flexibility of the system [1, 2]. In recent years, there has been an increasing interest among researchers with regard to underactuated systems due to their broad applications. Many benchmarks of underactuated systems have also emerged to validate various nonlinear control design techniques, such as Pendubot, Acrobot, inverted pendulum, Furuta Pendulum, TORA, VTOL (Vertical Take-Off and Landing) aircraft, etc. [3 –5].

The TORA or RTAC (Rotational-Translational ACtuator) system consists of a translational oscillation cart and an eccentric rotor attached to the cart. The system was originally studied as a simplified model of a dual-spin spacecraft to investigate the resonance capture phenomenon [6]. At that point it was considered to be a benchmark problem for nonlinear control [7]. For the system, both the translational position of the cart and the rotation angle of the rotor needed to be controlled. Since only the rotating angle is actuated, the TORA system is an underactuated system. Consequently, the system cannot be globally feedback linearized and the direct application of well-known nonlinear control schemes, such as feedback linearization, cannot guarantee its global stabilization. Jankovic et al. [8] proposed cascade-based control designs for the TORA system. Petres et al. [9] employed TORA to study the approximation and complexity trade-off capabilities of the tensor product distributed compensation-based control design. Bupp et al. [10] designed one integrator backstepping controller and three passive nonlinear controllers, and implemented them on an experimental testbed of TORA. Lee et al. [13] proposed an adaptive backstepping control scheme based on a wavelet-based neural network for the TORA system, and a compensated controller was provided to enhance the control performance. Avis et al. [11] presented an energy-based and entropy-based hybrid control framework for stabilization of the RTAC system, and compared the two controllers' performances through experiments. In [12], an equivalent-input-disturbance approach was developed to stabilize the TORA system with two steps. Moreover, stabilizing controllers with only rotor angle feedback were designed for the TORA system [14 –16]. Most of above results were based on simulations; however, the approaches in [10, 11, 13] were validated by experimental results.

Based on TORA, a novel 2-dimensional TORA (2DTORA) was proposed in [17]. The 2DTORA is an underactuated mechanical system that has one actuated rotor and two perpendicular translational oscillation carts instead of one. Compared with TORA, 2DTORA has one more configuration variable to be controlled while only the rotating angle of the rotor is actuated. Therefore, it is more difficult to design its stabilization controllers. To date there have been few studies focusing on the control of the 2DTORA system. In [17, 18], the dynamics, controllability, stability, and passivity of the 2DTORA system on a horizontal plane were studied and a PD controller based on passivity was designed to stabilize the system. According to the design techniques presented in [19], a Euler-Lagrange controller based on passivity of underactuated systems was designed to stabilize the 2DTORA system with only rotor angle feedback in [20].

Successful stabilizing control of the 2DTORA system demonstrates the possibility of controlling two perpendicular oscillations with one actuated rotation motion. This is an important inherent characteristic of the underactuated 2DTROA system which has potential applications to the active anti-vibration control of high buildings. It is well known that active anti-vibration control has better performance over passive anti-vibration control for the stability of building structures [21]. One of the disadvantages of active anti-vibration control for buildings is the cost since a full-actuation or a even redundant-actuation control system is needed [22]. However, based on the characteristic of the underactuated 2DTORA system, only one actuation is needed to control two perpendicular vibrations. Therefore, the idea of controlling the underactuated 2DTORA system could be applied to active anti-vibration control of building structures with a low cost.

However, there are no perfect horizontal planes in real applications. In other words, the motions of the 2DTORA system will be on a slope. In this paper, the 2DTORA system on a horizontal plane will be extended to the system on a slope. Dynamical modelling and equilibriums with controllability will be analysed in detail.

The rest of paper is organized as follows. In section 2, the dynamics of the 2DTORA is developed based on Lagrange equations. Next, the equilibriums and their controllability conditions are analysed in section 3. In section 4, simulation studies on the dynamics and controllability analysis are presented. Finally, conclusions are presented in the last section.

2. Dynamical Modelling

The 2DTORA system on the slope is depicted in Figure 1. The parameters of the system are defined as is the 2DTORA system on the horizontal plane in [17]. The outer oscillation cart of mass M_y is connected to a fixed wall by a linear spring of stiffness k_y; the inner cart of mass M_x is connected onto a wall of the outer cart by a linear spring of stiffness k_x. The outer cart and inner cart are constrained to have 1-dimensional translational motion with y and x denoting the travel distance respectively, and the two translational motions of the carts are perpendicular to each other. The actuated rotor attached to the inner cart has mass m and moment of inertia I about its centre of mass, which is located a distance r from the point about which the rotor rotates. In Figure 1, τ denotes the control torque applied to the rotor. Let x, $\dot{x}$ and y, $\dot{x}$ denote the translational position and velocity of the inner cart and outer cart, respectively. Define the rotor rotating away from the negative y-axis as the angular position θ and let $\dot{θ}$ denote the rotating velocity of the rotor. The motion occurs on a slope having an angle β with respect to the horizontal plane; therefore, gravitational forces have to be considered. Let g be the gravity constant and γ denote the heading angle of the 2DTORA system.

Figure 1.

2DTORA system on a slop

2.1 Modelling Based on Lagrange Equations

Since the system in Figure 1 is a multi-body dynamical system, Lagrange equations are employed to derive its dynamics. The general form of the Lagrange equations is

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{k}}) - \frac{\partial L}{\partial q_{k}} = Q_{k}

(1)

where L is the Lagrangian of the particle system; q_k is the general coordinate; k is number of the degree of freedom; Q_k denotes the general force. Dynamical modelling of the 2DTORA system will be processed by selecting x, y, θ as the general coordinates and τ as the general force.

The Lagrangian L is L = T − P, where T and P are the kinetic energy and potential energy of the system, respectively. The total kinetic energy T is a sum of kinetic energy T_x corresponding to the equivalent mass of M_x, kinetic energy T_y corresponding to the equivalent mass of M_y, and kinetic energy T_m of the rotor K_m. Therefore,

\begin{array}{l} T = T_{x} + T_{y} + T_{m} \\ = \frac{1}{2} (M_{x} + m) {\dot{x}}^{2} + \frac{1}{2} (M_{y} + m) {\dot{y}}^{2} \\ + m r \dot{θ} (\dot{x} \cos θ + \dot{y} \sin θ) + \frac{1}{2} (m r^{2} + I) {\dot{θ}}^{2} \end{array}

(2)

Let zero potential energy of the gravity be the centre of the carts when the springs are at free status. According to Figure 1, the arbitrary positions of the three mass points on the slope can be depicted as in Figure 2. The total potential energy is the sum of the energy saved in the two springs P_s and gravity energy P_g:

P = P_{s} + P_{g}

(3)

P_{s} = \frac{1}{2} k_{x} x^{2} + \frac{1}{2} k_{y} y^{2}

(4)

\begin{array}{l} P_{g} = - (M_{y} - M_{x}) g | o a | \sin β - M_{x} g | o b | \sin β - m g | o c | \sin β \\ = - (M_{y} - M_{x}) g y \sin γ \sin β \\ - M_{x} g [- x \sec γ + (y + x \tan γ) \sin γ] \sin β \\ - m g [- x \sec γ + (y + x \tan γ) \sin γ - r \sin (γ + θ)] \sin β \\ = - (M_{y} + m) g y \sin γ \sin β \\ + (M_{x} + m) g x \cos γ \sin β + m g r \sin (γ + θ) \sin β \end{array}

(5)

where |·| denotes the length of the line segment.

Figure 2.

Mass points analysis of the 2DTORA system on a slope

Figure 3.

Simulation results on dynamics with zero input and zero initial states: slope parameter γ = 30°, β = 60°; (left column) no friction, (right column) k_fx = k_fy = 0.05, k_fr = 0.03

Remark 1: For the system in Figure 1, the inner cart is a part of the outer cart. Therefore we always have M_y > M_x.

Remark 2: According to Figure 1, the mass points of the two degree oscillation of the carts are coincided at points o when the springs are at free status. However, as for the arbitrary position, the mass points of the two carts are not coincided, which has to be considered during the potential energy calculation. Based on Figure 2, as the mass point of outer cart moves away from the point o along the y axis, the mass point of inner cart moves away from the y axis along the x axis irrespective of the outer cart motion.

Remark 3: Normally, the angle β of the slope is caused by the assembly or inherent slope of the installation plane. Therefore, the slope angle is small. Here, without loss of generality, let 0° ≤ β < 90°. However, the heading angle γ denoting the slope direction corresponding to oscillation directions of the carts could be arbitrary. Therefore, let −180° < γ ≤ 180°.

Based on the selection of general coordinates and general force, the Lagrange equations of motion for the 2DTORA system on a slope are

\frac{d}{d t} (\frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial x} = - F_{x}

(6)

\frac{d}{d t} (\frac{\partial L}{\partial \dot{y}}) - \frac{\partial L}{\partial y} = - F_{y}

(7)

\frac{d}{d t} (\frac{\partial L}{\partial \dot{θ}}) - \frac{\partial L}{\partial θ} = τ - F_{r}

(8)

where F_x and F_y are the translational disturbance forces applied to the moving carts, and F_r is the disturbance torque applied to the rotor.

After some calculations, the dynamics of the 2DTORA system on a slope is

\begin{array}{l} (M_{x} + m) \ddot{x} + m r \cos θ \ddot{θ} - m r \sin θ {\dot{θ}}^{2} \\ + k_{x} x + (M_{x} + m) g \cos γ \sin β = - F_{x} \\ (M_{y} + m) \ddot{y} + m r \sin θ \ddot{θ} + m r \cos θ {\dot{θ}}^{2} \\ + k_{y} y - (M_{y} + m) g \sin γ \sin β = - F_{y} \\ m r \cos θ \ddot{x} + m r \sin θ \ddot{y} + \\ (m r^{2} + I) \ddot{θ} + m g r \cos (γ + θ) \sin β = τ - F_{r} \end{array}

Let q = [x, y, θ]^T, the above dynamics can be written in a compact form

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F = U

(9)

where M(q), C(q, $\dot{q}$ ), G(q), F, and U are the inertia matrix, Coriolis and centrifugal force matrix, potential energy matrix, disturbance force vector, and control input vector, respectively.

\begin{matrix} M (q) = [\begin{matrix} M_{x} + m & 0 & m r \cos θ \\ 0 & M_{y} + m & m r \sin θ \\ m r \cos θ & m r \sin θ & m r^{2} + I \end{matrix}], \\ C (q, \dot{q}) = [\begin{matrix} 0 & 0 & - m r \sin θ \dot{θ} \\ 0 & 0 & m r \cos θ \dot{θ} \\ 0 & 0 & 0 \end{matrix}], \\ G (q) = [\begin{matrix} k_{x} x + (M_{x} + m) g \cos γ \sin β \\ k_{y} y - (M_{y} + m) g \sin γ \sin β \\ m g r \cos (θ + γ) \sin β \end{matrix}], \\ F = [\begin{matrix} F_{x} \\ F_{y} \\ F_{r} \end{matrix}], U = [\begin{matrix} 0 \\ 0 \\ τ \end{matrix}] . \end{matrix}

Remark 4: From the dynamics equation (9), it is clear that q is the configuration variable vector of the system, θ is the actuated variable, and x, y are the unactuated variables. Since there are three configuration variables to be controlled with only one actuated variable, the 2DTORA system on a slope is an underactuated system.

2.2 Properties of the Dynamics

There are some general properties of these matrices. Note that M(q) is symmetric, and

\begin{matrix} \det (M) = M_{x} M_{y} m r^{2} + M_{x} m^{2} r^{2} \cos^{2} θ + M_{y} m^{2} r^{2} \sin^{2} θ \\ + (M_{x} m + M_{y} m + M_{x} M_{y} + m^{2}) I > 0 \end{matrix}

(10)

Therefore, M(q) is positive definite for all q. We also have

\dot{M} - 2 C = [\begin{matrix} 0 & 0 & m r \sin θ \dot{θ} \\ 0 & 0 & - m r \cos θ \dot{θ} \\ - m r \sin θ \dot{θ} & m r \cos θ \dot{θ} & 0 \end{matrix}]

(11)

which is a skew-symmetric matrix. This constitutes an important property

z_{} T (\dot{M} - 2 C) z = 0 \forall z ∊ ℝ^{3}

(12)

which can be used for establishing the passivity of the system. The potential energy of the system P is related to G(q) as

G = \frac{\partial P}{\partial q}

(13)

To establish the passivity of the system, we neglect the disturbance force matrix F in the dynamics of the 2DTORA (this is the same in the following unless otherwise stated).

The total energy of the system is given by

E = T + P = \frac{1}{2} {\dot{q}}^{T} M (q) \dot{q} + P

(14)

Differentiating E, based on (9) and (12) we obtain

\begin{array}{l} \dot{E} = {\dot{q}}^{T} M (q) \ddot{q} + \frac{1}{2} {\dot{q}}^{T} \dot{M} (q) \dot{q} + {\dot{q}}^{T} G (q) \\ = {\dot{q}}^{T} [- C (q, \dot{q}) \dot{q} - G (q) + U + \frac{1}{2} \dot{M} (q) \dot{q}] + {\dot{q}}^{T} G (q) \\ = {\dot{q}}^{T} U = \dot{θ} τ \end{array}

(15)

Integrating both sides of the above equation we get

\int_{0}^{t} \dot{θ} (t) τ d t = E (t) - E (0) \geq - E (0)

(16)

Therefore, the system having τ as input and $\dot{θ}$ as output is passive.

Remark 5: For the dynamics equation (9), when the slope angle is zero, i.e., β = 0, the dynamics becomes the dynamics on a horizontal plane. By neglecting the friction torque of rotor F_r, Eq. (9) is

(M_{x} + m) \ddot{x} + m r \cos θ \ddot{θ} - m r \sin θ {\dot{θ}}^{2} + k_{x} x = - F_{x}

(17)

(M_{y} + m) \ddot{y} + m r \sin θ \ddot{θ} + m r \cos θ {\dot{θ}}^{2} + k_{y} y = - F_{y}

(18)

m r \cos θ \ddot{x} + m r \sin θ \ddot{y} + (m r^{2} + I) \ddot{θ} = τ

(19)

which is the dynamics derived for the 2DTORA system on a horizontal plane developed in [17]. Moreover, if we remove one of the translational motions from the above dynamics, it is easy to get the dynamics of the TORA. As we remove the motion in y axis, the dynamics becomes as it is in the TORA system used in [7]

(M_{x} + m) \ddot{x} + m r \cos θ \ddot{θ} - m r \sin θ {\dot{θ}}^{2} + k_{x} x = - F_{x}

(20)

m r \cos θ \ddot{x} + (m r^{2} + I) \ddot{θ} = τ

(21)

which is a benchmark nonlinear system.

3. Equilibriums and their Controllability

Defining x = [x₁, x₂, x₃, x₄, x₅, x₆]^T = [x, $\dot{x}$ , y, $\dot{y}$ , θ, $\dot{θ}$ ]^T, we can rewrite the system (9) in a general affine form

\dot{x} = f (x) + g (x) u

(22)

where u = τ. Let

a_{1} = M_{y} + m, a_{2} = M_{x} + m, b = m r, c = m r^{2} + I

(23)

we have

f (x) = [\begin{matrix} x_{2} \\ \frac{f_{2}}{\det (M)} \\ x_{4} \\ \frac{f_{4}}{\det (M)} \\ x_{6} \\ \frac{f_{6}}{\det (M)} \end{matrix}], g (x) = [\begin{matrix} 0 \\ \frac{- a_{1} b \cos x_{5}}{\det (M)} \\ 0 \\ \frac{- a_{2} b \sin x_{5}}{\det (M)} \\ 0 \\ \frac{a_{1} a_{2}}{\det (M)} \end{matrix}]

(24)

where

\begin{array}{l} f_{2} = - (a_{1} c - b^{2} \sin^{2} x_{5}) k_{x} x_{1} - b^{2} \sin x_{5} \cos x_{5} k_{y} x_{3} \\ + (M_{y} c + m I) b \sin x_{5} x_{6}^{2} + a_{1} b^{2} g [\sin x_{5} \cos x_{5} \sin γ \\ + \cos x_{5} \cos (γ + x_{5})] \sin β \\ - (a_{1} c - b^{2} \sin^{2} x_{5}) a_{2} g \cos γ \sin β \\ f_{4} = - b^{2} \sin x_{5} \cos x_{5} k_{x} x_{1} - (a_{2} c - b^{2} \cos^{2} x_{5}) k_{y} x_{3} \\ - (M_{x} c + m I) b \cos x_{5} x_{6}^{2} - a_{2} b^{2} g [\sin x_{5} \cos x_{5} \cos γ \\ - \sin x_{5} \cos (γ + x_{5})] \sin β \\ + (a_{2} c - b^{2} \cos^{2} x_{5}) a_{1} g \sin γ \sin β \\ f_{6} = a_{1} b \cos x_{5} k_{x} x_{1} + a_{2} b \sin x_{5} k_{y} x_{3} \\ - (M_{y} - M_{x}) b^{2} \sin x_{5} \cos x_{5} x_{6}^{2} \end{array}

To get the equilibriums of the system, let u = 0 and f(x) = 0. Thus x₂ = x₄ = x₆ = 0, and then f₂ = 0, f₄ = 0, f₆ = 0 can be simplified as

\begin{matrix} 0 = - (a_{1} c - b^{2} \sin^{2} x_{5}) k_{x} x_{1} - b^{2} \sin x_{5} \cos x_{5} k_{y} x_{3} + \\ (a_{1} b^{2} \cos^{2} x_{5} + a_{2} b^{2} \sin^{2} x_{5} - a_{1} a_{2} c) g \cos γ \sin β \end{matrix}

(25)

\begin{matrix} 0 = - b^{2} \sin x_{5} \cos x_{5} k_{x} x_{1} - (a_{2} c - b^{2} \cos^{2} x_{5}) k_{y} x_{3} + \\ (a_{1} a_{2} c - a_{1} b^{2} \cos^{2} x_{5} - a_{2} b^{2} \sin^{2} x_{5}) g \sin γ \sin β \end{matrix}

(26)

0 = a_{1} b \cos x_{5} k_{x} x_{1} + a_{2} b \sin x_{5} k_{y} x_{3}

(27)

Substituting (27) into (25) and (26), we have

a_{0} \frac{k_{x} x_{1}}{a_{2}} = - a_{0} g \cos γ \sin β

(28)

a_{0} \frac{k_{y} x_{3}}{a_{1}} = a_{0} g \sin γ \sin β

(29)

where

a_{0} = a_{1} a_{2} c - a_{1} b^{2} \cos^{2} x_{5} - a_{2} b^{2} \sin^{2} x_{5} = \det (M) > 0

Because the parameters k_x, k_y, a₁, a₂, a₀, g are all positive, we have the equilibriums of the cart positions

x_{1} = - \frac{a_{2} g \cos γ \sin β}{k_{x}}

(30)

x_{3} = \frac{a_{1} g \sin γ \sin β}{k_{y}}

(31)

Substituting the equilibriums of the cart positions into (27), we have

a_{1} a_{2} b g \cos (γ + x_{5}) \sin β = 0

(32)

Because the parameters a₁, a₂, b, g are all positive, we have the equilibriums of the rotor angle

{\begin{array}{l} x_{5} = θ_{e}, β = 0 \\ x_{5} = (\frac{2 k + 1}{2}) π - γ, k = 0, \pm 1, \pm 2, \dots,0 < β {< 90}^{\circ} \end{array}

(33)

where θ_e is an arbitrary angle.

Combining the equilibriums of the cart positions and rotor angle, the equilibriums of the 2DTORA system on a slope are

x_{e} ​​ = ​​ [\frac{- a_{2} g \cos γ \sin β}{k_{x}},0, \frac{a_{1} g \sin γ \sin β}{k_{y}}, (\frac{2 k + 1}{2}) π - γ,0]

(34)

where k = 0, ± 1, ±2, ···.

Remark 6: When the slope angle is zero, i.e., sin β = 0, based on (33) and (34), the equilibriums of the 2DTORA system on a slope become its equilibriums on a horizontal plane x_e = [0, 0, 0, 0, θ_e, 0]^T. This is in accordance with the equilibrium result developed in [18].

Remark 7: The 2DTORA system has two types of equilibriums, i.e., stable equilibriums and unstable equilibriums. When the system is on a horizontal plane, all its equilibriums are stable and the potential energy of gravity is constant for all equilibriums. However, for the system on a slope, the unstable and stable equilibriums are as follows, respectively

x_{e_{1}} ​​ = ​​ [\frac{- a_{2} g \cos γ \sin β}{k_{x}},0, \frac{a_{1} g \sin γ \sin β}{k_{y}},0, \frac{(4 k + 1) π}{2} - γ,0]

(35)

x_{e_{2}} ​​ = ​​ [\frac{- a_{2} g \cos γ \sin β}{k_{x}},0, \frac{a_{1} g \sin γ \sin β}{k_{y}},0, \frac{(4 k + 3) π}{2} - γ,0]

(36)

It is straightforward that the system has smallest potential energy of gravity at the stable equilibriums.

Once the equilibriums of the 2DTORA system are obtained, one wants to know if all the equilibriums in Eq. (34) are controllable or controllable under what condition.

We know that the system in Eq. (22) is controllable or reachable about its equilibriums if the following condition is fulfilled:

\dim 〈 a d_{f} | R (f) 〉 (x_{e}) = 6

(37)

which corresponds to the full rank condition for the following matrix

C = [B, A B, A^{2} B, A^{3} B, A^{4} B, A^{5} B]

(38)

where

\begin{array}{l} A : = (\partial / \partial x) f (x_{e}) \\ B : = g (x_{e}) \end{array}

The matrices A, B and C can be calculated as

\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ a_{21} & 0 & a_{23} & 0 & a_{25} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ a_{41} & 0 & a_{43} & 0 & a_{45} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ a_{61} & 0 & a_{63} & 0 & a_{65} & 0 \end{matrix}], B = [\begin{matrix} 0 \\ b_{2} \\ 0 \\ b_{4} \\ 0 \\ b_{6} \end{matrix}], \\ C = [\begin{matrix} 0 & b_{2} & 0 & c_{14} & 0 & c_{16} \\ b_{2} & 0 & c_{23} & 0 & c_{25} & 0 \\ 0 & b_{4} & 0 & c_{34} & 0 & c_{36} \\ b_{4} & 0 & c_{43} & 0 & c_{45} & 0 \\ 0 & b_{6} & 0 & c_{54} & 0 & c_{56} \\ b_{6} & 0 & c_{63} & 0 & c_{65} & 0 \end{matrix}] \end{matrix}

where

\begin{array}{l} a_{21} = - (a_{1} c - b^{2} \sin^{2} θ_{e}) k_{x} {[\det M]}^{- 1}, \\ a_{23} = - b^{2} \sin θ_{e} \cos θ_{e} k_{y} {[\det M]}^{- 1}, \\ a_{25} = - a_{1} b^{2} g \sin (2 θ_{e} + γ) \sin β {[\det M]}^{- 1}, \\ a_{41} = - b^{2} \sin θ_{e} \cos θ_{e} k_{x} {[\det M]}^{- 1}, \\ a_{43} = - (a_{2} c - b^{2} \cos^{2} θ_{e}) k_{y} {[\det M]}^{- 1}, \\ a_{45} = a_{2} b^{2} g \cos (2 θ_{e} + γ) \sin β {[\det M]}^{- 1}, \\ a_{61} = a_{1} b \cos θ_{e} k_{x} {[\det M]}^{- 1}, \\ a_{63} = a_{2} b \sin θ_{e} k_{y} {[\det M]}^{- 1}, \\ a_{65} = a_{1} a_{2} b g \sin (θ_{e} + γ) \sin β {[\det M]}^{- 1}, \\ b_{2} = - a_{1} b \cos θ_{e} {[\det M]}^{- 1}, \\ b_{4} = - a_{2} b \sin θ_{e} {[\det M]}^{- 1}, \\ b_{6} = a_{1} a_{2} {[\det M]}^{- 1}, \\ c_{23} = c_{14} = a_{21} b_{2} + a_{23} b_{4} + a_{25} b_{6}, \\ c_{43} = c_{34} = a_{41} b_{2} + a_{43} b_{4} + a_{45} b_{6}, \\ c_{63} = c_{54} = a_{61} b_{2} + a_{63} b_{4} + a_{65} b_{6}, \\ c_{25} = c_{16} = a_{21} c_{23} + a_{23} c_{43} + a_{25} c_{63}, \\ c_{45} = c_{36} = a_{41} c_{23} + a_{43} c_{43} + a_{45} c_{63}, \\ c_{65} = c_{56} = a_{61} c_{23} + a_{63} c_{43} + a_{65} c_{63} . \end{array}

Based on the property on determinant of a matrix, we have

| C | = | \begin{matrix} b_{2} & c_{23} & c_{25} & 0 & 0 & 0 \\ b_{4} & c_{43} & c_{45} & 0 & 0 & 0 \\ b_{6} & c_{63} & c_{65} & 0 & 0 & 0 \\ 0 & 0 & 0 & b_{2} & c_{14} & c_{16} \\ 0 & 0 & 0 & b_{4} & c_{34} & c_{36} \\ 0 & 0 & 0 & b_{6} & c_{54} & c_{56} \end{matrix} | = | \begin{matrix} C'_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & C'_{3 \times 3} \end{matrix} |

(39)

Therefore, the condition of full rank of the matrix C is transformed to the full rank condition of the matrix C'. Here, we calculate the determinant det(C') ≠ 0 to find out its full rank conditions.

\begin{matrix} \det (C^{'}) = \frac{- a_{0} b^{2}}{2 [\det M]^{6}} (a_{2} k_{y} - a_{1} k_{x}) \\ \begin{matrix} (- a_{1} a_{2}^{2} b [\cos θ_{e} \sin (γ + θ_{e}) - g \sin (2 θ_{e} + γ)] \sin β - k_{x} \cos θ_{e} a_{0}) \\ (a_{1}^{2} a_{2} b [- \cos γ + (1 + 2 g) \cos (2 θ_{e} + γ)] \sin β - 2 k_{y} \sin θ_{e} a_{0}) \end{matrix} \end{matrix}

(40)

The conditions when the above equation equals to zero need to be discussed.

Case 1: If the 2DTORA system is on the horizontal plane, i.e., β = 0, the determinant of matrix C′ becomes

\det (C') = \frac{- b^{2} k_{x} k_{y} \sin θ_{e} \cos θ_{e}}{{[\det (M)]}^{3}} (a_{2} k_{y} - a_{1} k_{x})

(41)

where a₀, a₁, a₂, b, k_x, k_y, det(M) are all positive. Therefore, the full rank conditions for matrix C' (also C) are

\sin θ_{e} \neq 0

(42)

\cos θ_{e} \neq 0

(43)

(M_{x} + m) k_{y} - (M_{y} + m) k_{x} \neq 0

(44)

which lead to

θ_{e} \neq k \frac{π}{2}, k = 0, \pm 1, \pm 2, \dots

(45)

\frac{M_{x} + m}{k_{x}} \neq \frac{M_{y} + m}{k_{y}}

(46)

Condition (45) shows that when the rotor angles are aligned to the oscillation directions of the carts, the system is uncontrollable. Condition (46) shows that when the oscillation periods of the two directions of the cart-spring system are the same, the system is uncontrollable.

Remark 8: For a mass-spring system consisting of mass M and spring constant K, its oscillation period is $T = 2 π \sqrt{M / K}$ .

Case 2: If the 2DTORA system is on a slope, i.e., 0 < β < 90°, for the stable equilibriums (36), i.e., θ_e = (2k + 1.5) π − γ, the determinant of matrix C' becomes

\begin{array}{l} \det (C') = \frac{b^{2} \sin θ_{e} \cos θ_{e}}{{[\det (M)]}^{5}} (a_{2} k_{y} - a_{1} k_{x}) \\ [a_{1} a_{2}^{2} b (g - 1) \sin β + k_{x} a_{0}] [a_{1}^{2} a_{2} b (g + 1) \sin β - k_{y} a_{0}] \\ = \frac{b^{2} \sin γ \cos γ}{{[\det (M)]}^{5}} (a_{2} k_{y} - a_{1} k_{x}) \\ [a_{1} a_{2}^{2} b (g - 1) \sin β + k_{x} {a^{'}}_{0}] [a_{1}^{2} a_{2} b (g + 1) \sin β - k_{y} {a^{'}}_{0}] \end{array}

(47)

where a₁, a₂, b, k_x, k_y, det(M) are all positive and

\begin{array}{r} a_{1} a_{2}^{2} b (g - 1) \sin β + k_{x} a_{0} > 0 \\ {a^{'}}_{0} = M_{x} M_{y} b r + M_{x} b^{2} \sin^{2} γ + M_{y} b^{2} \cos^{2} γ + a_{1} a_{2} I > 0 \end{array}

Therefore, for the stable equilibriums of the 2DTORA system on a slope, the controllable equilibriums and their controllable conditions are

γ \neq k \frac{π}{2}, k = 0, \pm 1, \pm 2, \dots

(48)

\frac{M_{x} + m}{k_{x}} \neq \frac{M_{y} + m}{k_{y}}

(49)

k_{y} {a^{'}}_{0} \neq a_{1}^{2} a_{2} b (g + 1) \sin β

(50)

Based on the relationship between θ and γ, for the stable equilibriums of the 2DTORA system on a slope, conditions (48) and (49) are the same. For condition (50), it can be rewritten as

\begin{array}{l} M_{x} M_{y} m r^{2} + M_{x} m^{2} r^{2} \sin^{2} γ + M_{y} M_{x} m^{2} r^{2} \cos^{2} γ \\ + (M_{x} + m) (M_{y} + m) I \\ \neq k_{y}^{- 1} (M_{x} + m) (M_{y} + m)^{2} m r (g + 1) \sin β \end{array}

One can see that the condition (50) is a constrain of nine physical parameters M_x, M_y, m, r, I, k_y, γ, β, g. This condition does not like conditions (48) and (49). For condition (48), people may want to stabilize the rotor angle to these special angles; and for condition (49), some symmetrical plants could have the same oscillation periods in x and y direction. However, condition (50) could never happen in practice. In other words, condition (50) can always be taken as fulfilled in reality.

Case 3: If the 2DTORA system is on a slope, i.e., 0 < β < 90°, for the unstable equilibriums (35), i.e., θ_e = (2k + 0.5) π − γ, the determinant of matrix C′ becomes

\begin{array}{l} \det (C') = \frac{- b^{2} \sin θ_{e} \cos θ_{e}}{{[\det (M)]}^{5}} (a_{2} k_{y} - a_{1} k_{x}) \\ [a_{1} a_{2}^{2} b (1 - g) \sin β + k_{x} a_{0}] [a_{1}^{2} a_{2} b (1 + g) \sin β + k_{y} a_{0}] \\ = \frac{- b^{2} \sin γ \cos γ}{{[\det (M)]}^{5}} (a_{2} k_{y} - a_{1} k_{x}) \\ [a_{1} a_{2}^{2} b (1 - g) \sin β + k_{x} {a^{'}}_{0}] [a_{1}^{2} a_{2} b (1 + g) \sin β + k_{y} {a^{'}}_{0}] \end{array}

where a′₀, a₁, a₂, b, k_x, k_y, det(M) are all positive and

a_{1}^{2} a_{2} b (1 + g) \sin β + k_{y} {a^{'}}_{0} > 0

(51)

Therefore, for the unstable equilibriums of the 2DTORA system on a slope, the controllable equilibriums and their controllable conditions are

γ \neq k \frac{π}{2}, k = 0, \pm 1, \pm 2, \dots

(52)

\frac{M_{x} + m}{k_{x}} \neq \frac{M_{y} + m}{k_{y}}

(53)

k_{x} {a^{'}}_{0} \neq a_{1} a_{2}^{2} b (g - 1) \sin β

(54)

Comparing the above conditions for the unstable equilibriums of the 2DTORA system on a slope with the stable ones (48), (49), and (50), they are similar. However, the unstable equilibriums are not focused upon in most research, especially for applications in the active anti-vibration control of buildings.

As a conclusion, for the 2DTORA system on a horizontal plane or slope, the controllable equilibriums and their main controllable conditions are: (1) the rotor angle cannot be set to the directions aligned to the oscillation directions of the carts as formulated by (45) or (48); (2) the oscillation periods of the two directions of the cart-spring system cannot be the same.

Remark 9: For all three configuration variables x, y and θ of the system, the actuated variable θ can be controlled directly by the input torque; however, the unactuated degrees of freedom of the cart oscillation must be controlled through system coupling. The controllable equilibriums can be explained according to the dynamics (9) intuitively: the translation position x is controlled by the input torque τ with a coupling coefficient related to cosθ. If the destination θ is set to $\frac{2 k + 1}{2} π$ , where k is an integer and the θ can always be stabilized to $\frac{2 k + 1}{2} π$ with a controller, then we have cos θ = 0 which means that the nonlinear coupling to control the translational motion of the cart in x direction will disappear. In this case, the input control torque τ will lose control of the cart position in x direction after the θ is brought to $\frac{2 k + 1}{2} π$ by the control torque τ. Similar arguments can be applied to y direction.

4. Simulation Study

This section validates the dynamics and analysis on the equilibriums and their controllability of the 2DTORA system on a slope through simulations based on Simulink™. In the following simulations, the physical parameters of the 2DTORA system are based on [7]: M_x=1.3608kg, M_y=2.7216kg, m=0.096kg, k_x=186.3N/m, k_y=279.5N/m, r=0.0592m, I=0.0002175kgm², g=9.8 m/s².

Firstly, simulation studies are performed to validate the equilibriums in Eq. (34) based on the system dynamics in Eq. (9). In the simulations, the initial state of the system is set to zero [x, $\dot{x}$ , y, $\dot{y}$ , θ, $\dot{θ}$ ]^T = [0, 0, 0, 0, 0, 0]^T and no control input τ = 0. The slope parameters are set as γ = 30°, β = 60°. To show the equilibriums clearly in the simulation curves and compare the performance of system dynamics without and with disturbance force, we assume the disturbance friction forces as F_x = k_fx $\dot{x}$ , F_y = k_fy $\dot{y}$ , F_r = k_fr $\dot{θ}$ , where k_fx = k_fy = 0.005, k_fr = 0.03 are the friction constants. The simulation results are shown in Figure 3 where its left column is without friction and the right column is with friction.

According to the left column of the simulation results in Figure 3, for the 2DTORA system on a slope without friction, carts oscillate along x-axis and y-axis and the oscillation centres are the equilibriums. Moreover, the oscillation amplitudes of x-axis and y-axis are complementary because the overall system energy is conservative. The curve of rotor angle θ varies without obvious feature, which is a combination of rounds of rotation and oscillation within a round based on its initial condition. From the right column of the simulation results in Figure 3, when the frictions are ejected into the system, the cart motion of x-axis and y-axis is damped oscillation with respect to their equilibriums x=−0.0575m, y=0.0428m. The rotor angle will stick to its equilibrium and oscillate around it until damped to θ = −120° under friction. Without control input, the simulation results show that: in the no friction case, the cart will oscillate around its equilibriums and rotor rotate without obvious features under a non-stable initial condition; in the friction case, the cart and rotor will damped oscillate to their equilibriums.

The second simulation studies were performed to validate the controllable equilibriums, i.e., that the rotor angle cannot be set to the oscillation directions of the carts. We arranged the slope parameters and initial rotor angle to guarantee that the initial rotor angle is with its stable equilibrium and aligned to the oscillation directions of x-axis and y-axis. The simulation results with disturbance forces that have constants as k_fx = k_fy = 0.05 and k_fr = 0.03 are shown in Figure 4.

Figure 4.

Simulation results on the dynamics with zero input and k_fx = k_fy = 0.05, k_fr = 0.03: (left column) slope parameter γ = 0°, β = 60°, initial state [0, 0, 0, 0, $- \frac{π}{2}$ , 0]^T; (right column) slope parameter γ = 90°, β = 60°, initial state [0, 0, 0, 0, π, 0]^T

Figure 5.

Simulation results on dynamics with zero initial states and no friction: slope parameter γ = −30°, β = 9°; (left column) θ_e = −60°, (right column) θ_e = −60° and (M_x + m)k_y = M_y + m)k_x

The initial condition of the left column in Figure 4: slope parameters γ = 0°, β = 60°, initial state [x, $\dot{x}$ , y, $\dot{y}$ , θ, $\dot{θ}$ ]^T = [0, 0, 0, 0, $- \frac{π}{2}$ , 0]^T. These initial conditions guarantee that the initial angle of the rotor is its stable equilibrium and aligned to the x-axis oscillation motion of the cart. One can see that the x-axis cart is damped to its equilibrium from its initial condition under friction. However, the y-axis cart and the rotor are kept with the initial stable equilibriums. This shows that the oscillation of cart of x-axis has no effect on the rotor rotation when the rotor is set to the direction of x-axis. Symmetrically, the control input τ will have no effect on the oscillation motion of the x-axis cart when the rotor angle is finalized to the angle aligned to oscillation direction of the x-axis cart. The reason for this is that the dynamical coupling between rotor rotation and oscillation of the x-axis disappears since cos θ = 0. The initial condition of the right column in Figure 4: slope parameters γ = 90°, β = 60°, initial state [x, $\dot{x}$ , y, $\dot{y}$ , θ, $\dot{θ}$ ]^T = [0, 0, 0, 0, −π, 0]^T. These initial conditions guarantee that the initial angle of the rotor is its stable equilibrium and aligned to the y-axis oscillation motion of the cart. One can see that the y-axis cart is damped to its equilibrium from its initial condition under friction. However, the x-axis cart and the rotor are kept with the initial stable equilibriums. This shows that the oscillation of the cart of the y-axis has no effect on the rotor rotation when the rotor is set to the direction of y-axis. Symmetrically, the control input τ will have no effect on the oscillation motion of the y-axis cart when the rotor angle is finalized to the angle aligned to oscillation direction of the y-axis cart. The reason for this is that the dynamical coupling between rotor rotation and oscillation of the y-axis disappears since sin θ = 0.

Finally, simulation studies are performed to validate the controllable condition of the equilibriums, i.e., the oscillation periods of the carts in two directions cannot be the same. To proceed with this simulation study, a controller for the system is needed. Based on the control design for the 2DTORA system on a horizontal plane in [18], a similar Lyapunov function for the 2DTORA system on a slope can be designed as

V (q, \dot{q}) = k_{1} (E - E_{0}) + \frac{k_{2}}{2} {(θ - θ_{0})}^{2}

(55)

where k₁, k₂ are positive constant, and

\begin{array}{l} E_{0} ​​ = ​​ - \frac{1}{2} k_{x}^{- 1} a_{2}^{2} g^{2} \cos^{2} γ \sin^{2} β - \frac{1}{2} k_{y}^{- 1} a_{2}^{2} g^{2} \sin^{2} γ \sin^{2} β \\ - b g \sin β \\ θ_{0} = k π + \frac{3}{2} π - γ \end{array}

are system energy and typical rotor angle θ ∊ (−π, π] at stable equilibriums. Since the smallest system energy is E₀, the designed control Lyapunov function (55) is positive definite. Based on the passivity property (16), the stabilizing control input can be yielded similarly as in [18]

τ = - \frac{1}{k_{1}} [k_{2} (θ - θ_{0}) + k_{3} \dot{θ}]

(56)

where k₃, > 0.

The simulation results of the close-loop control system consisting of dynamics (9) without friction and controller (56) are shown in Figure 5. In the simulations, the parameters are set as: slope parameters γ = −30°, β = 9°, initial state [x, $\dot{x}$ , y, $\dot{y}$ , θ, $\dot{θ}$ ]^T = [0, 0, 0, 0, 0, 0]^T, controller parameters k₁ = 30, k₂ = 2, k₃ = 0.12, destination rotor angle θ = −60°. The left column of Figure 5 is the simulation results without changing of the physical parameters of the 2DTAORA system, while the right column of Figure 5 is the simulation results with a new spring constant in the y-axis oscillation k_y=360.3N/m, i.e., k_y = (M_y + m)k_x/(M_x + m) which fulfils that oscillation periods of the carts in two directions are the same. Based on the simulation results of the left column in Figure 5, it is shown that the controller (56) can stabilize the 2DTORA system from its initial condition to system equilibrium [−0.0104,0, −0.0077,0, −60, 0]^T in 60s. Based on the simulation results of right column in Figure 5, one can see that once the oscillation periods of the two carts are the same, both cart movements are uncontrollable although the rotor can be brought to its destination angle in 20s. The simulation results have verified the analysis in the last section successfully.

5. Conclusions

This paper presents the dynamics and analysis of the equilibriums and their controllable conditions of the 2DTORA systems on slopes. It shows that: (1) stable equilibriums of the system are located at the equilibriums with the smallest potential energy of gravity; (2) the rotor angle cannot be set to the directions aligned to the oscillation directions of the carts; (3) the oscillation periods of the two directions of the cart-spring system cannot be the same. The simulation results based on Simulink™ have verified the correctness and feasibility of the developed dynamics and controllability analysis.

The dynamics of the 2DTORA system on slopes is an extension of the dynamics of the 2DTORA system on a horizontal plane. Straightforwardly, it can be reduced to the dynamics of the TORA system on a slop or eventually the TORA system on a horizontal plane which serves as a benchmark nonlinear system. Since the motions of the 2DTORA system cannot be on a perfect horizontal plane in practical applications with gravity, the developed dynamics and related analysis in this paper are more general.

Footnotes

6.

The first author would like to thank the financial support from the National Science Foundation of China (NSFC) under grant no. 11102039 and the Natural Science Foundation of Jiangsu Province under grant no. BK2011456.

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