Dynamical Analysis and Stabilizing Control of Inclined Rotational Translational Actuator Systems

Rotational translational actuator (RTAC) system, whose motions occur in horizontal planes, is a benchmark for studying of control techniques.This paper presents dynamical analysis and stabilizing control design for theRTACsystemon a slope. Based onLagrange equations, dynamics of the inclined RTAC system is achieved by selecting cart position and rotor angle as the general coordinates and torque acting on the rotor as general force. The analysis of equilibriums and their controllability yields that controllability of equilibriums depends on inclining direction of the inclined RTAC system. To stabilize the system to its controllable equilibriums, a proper control Lyapunov function including system energy, which is used to show the passivity property of the system, is designed. Consequently, a stabilizing controller is achieved directly based on the second Lyapunov stability theorem. Finally, numerical simulations are performed to verify the correctness and feasibility of our dynamical analysis and control design.


Introduction
Underactuated systems are a class of mechanical control systems whose control inputs are less than the number of configuration variables [1][2][3].The RTAC system consisting of an unactuated translational oscillation cart and an actuated eccentric rotor attached to the cart is originally studied as a simplified model of a dual-spin spacecraft to investigate the resonance capture phenomenon [4].Then, it is brought to us as a benchmark problem for nonlinear control by Bupp et al. [5].The control objectives of RTAC system are to stabilize the translational position of the cart around its equilibrium and regulate the angular position of the rotor around one of its multiple equilibriums with the only control input torque acting on the rotor.
Stabilizing control design of RTAC has been extensively studied.Jankovic et al. [6] proposed cascade-based control designs for RTAC system.Bupp et al. [7] designed one integrator backstepping controller and three passive nonlinear controllers and implemented them on an experimental test bed of RTAC. Lee and Chang [8] proposed an adaptive backstepping control scheme based on a wavelet-based neural network for RTAC system, and a compensated controller has been provided to enhance the control performance.Tsiotras et al. [9] applied the theory of  2 disturbance attenuation for RTAC system.Petres et al. [10] employed RTAC to study the approximation and complexity tradeoff capabilities of the tensor product distributed compensation based control design.In [11], an equivalent-input-disturbance approach was developed to stabilize RTAC system with two steps based on the state variables of position.Moreover, in [12][13][14], controllers with only rotor angle feedback were designed for RTAC system.Most of the above results were based on simulation; however, approaches in [7,8] had been validated through experimental results.
RTAC system considered in all the above mentioned papers is on horizontal planes; therefore, no gravity effect is considered in the dynamics and control design.Gao [15] considered dynamics and energy based control of a RTAC system with rotating motion appearing in a vertical plane, where gravity affecting the rotor motion has to be included.Avis et al. [16] presented an energy-based and entropy-based hybrid control framework to stabilize RTAC system with rotating motion in a vertical plane and compared the two controllers' performance with experiments.However, it should be noted that, in real applications, absolute horizontal or vertical plane does not exist.In other words, the motions of RTAC system will be in inclined planes because of the assembly or inherent unlevel installation plane.In this paper, dynamical modeling and equilibriums' controllability of inclined RTAC systems will be analyzed in detail.And according to the passivity of the system, an energy-based control technique [15,[17][18][19][20] is employed to design a simple PD (proportional derivative) controller for the inclined RTAC system.The rest of paper is organized as follows.Dynamics of the inclined RTAC is developed according to Lagrange equations in Section 2. Based on the developed dynamics, its equilibriums and their controllability conditions are analyzed in Section 3.After that, a proper Lyapunov function including system energy is designed to generate a simple controller for the system in Section 4. Simulation results and discussions are provided in Section 5.And conclusions are summarized in the last section.

Dynamics
According to the benchmark system RTAC on the horizontal plane, inclined RTAC system is depicted as in Figure 1.A coordinate frame  is attached to the base of the RTAC system, with the origin at the center of the cart when the spring is with free status.The -axis is along the translational movement of the cart, and the -axis is perpendicular to the base plane of the RTAC system.According to the definition of , another world frame  is defined, as shown in Figure 1, for the slope on which the RTAC system locates.The motions of RTAC system occur in the inclined plane  having an angle  with respect to the horizontal plane , and inclining direction of the RTAC system is denoted with  as shown in Figure 1.Parameters of the inclined RTAC system are defined similarly as in [5] according to the coordinate frame .The cart of mass   is connected to a fixed base by a linear spring of stiffness   .The cart is constrained to have one-dimensional translational motion with  denoting the travel distance.The actuated rotor attached to the cart has mass  and moment of inertia  about its center of mass, and the eccentric distance of the rotor is .Control input torque applied to the rotor is denoted by .Let  and ẋ denote the translational position and velocity of the cart, respectively, and let  and θ denote angular position and its velocity of the rotor rotating away from the negative -axis, respectively.Since the gravitational force has to be considered for the inclined RTAC system, let  be the gravity constant.
The total kinetic energy  of the system is the sum of kinetic energy   , corresponding to the equivalent mass of translational cart and kinetic energy   of the rotational rotor: To calculate the potential energy of the system, let zero potential energy of the gravity be the center of the cart when the spring is with free status, that is, the origin of the coordinates frame .According to Figure 1, the mass points of cart  1 and rotor  on the slope can be depicted as in Figure 2, where  ⊥ ,  1  1 ‖,  1  1  2 ‖, and  is the projection of point  on line .The planar coordinates  are extracted from the coordinates frame  in Figure 1.The circle with center  1 and radius  denotes the rotating trajectory of the rotor.Because, at point , the system has zero potential energy both for the spring and the gravity, the total potential energy, which is the sum of the potential energy   corresponding to the spring and potential energy   corresponding to the gravity, can be calculated as where | ⋅ | means the length of the line segment.Therefore, the Lagrangian  can be calculated as By choosing  and  as the generalized coordinates and  as the generalized force, the Lagrange equations of motion for the inclined RTAC system are as follows: where   and   denote the disturbance inputs acting on the translational moving cart and rotating rotor, respectively.Dynamics of the system can be calculated as By choosing the variable vector q = (, ), the above two equations can be written in a compact form as follows: where M(q), C(q, q ), G(q), N, and U represent inertia matrix, Coriolis and centrifugal force matrix, potential matrix, disturbance force vector, and control input vector of the inclined RTAC system, respectively, Remark 1.It can be observed straightforwardly from dynamics (8) that  is the actuated variable and  is the unactuated variable in the configuration variable vector q.Because the unactuated variable  has to be stabilized around its equilibrium and actuated variable  has to be regulated around one of its equilibriums with the only control input , the inclined RTAC system is an underactuated system.
Remark 2. The inclined angle  of the system could be caused by assembly or unlevel installation plane.Without loss of generality, it can be defined as  ∈ [0 ∘ , 90 ∘ ].However, the angle  denoting inclining direction of the system could be arbitrary.Therefore, it can be selected as  ∈ [0 ∘ , 360 ∘ ).
Remark 3. When the inclined angle  is zero, that is, the RTAC system on a horizontal plane, dynamics of inclined RTAC system (8) can be deduced as (neglecting the friction torque of rotor   ) which is the same as the dynamics of RTAC system provided in [5].

Analysis of Controllable Equilibriums
To derive the equilibriums of the system, we neglect the disturbance force matrix N in the dynamics of the inclined RTAC system.By choosing the system state x = ( 1 ,  2 ,  3 ,  4 ) = (, ẋ , , θ ) and system input  = , the dynamics (8) can be rewritten in general affine form as follows: where To solve the equilibriums of the unforced system, let  = 0 and ẋ = 0. Therefore, we have  2 =  4 = 0 and  2 =  4 = 0 which can be expressed as Simplifying the above two equations by considering that all the parameters   , , , , and  are positive, we have det (M)  cos ( +  3 ) sin  = 0. ( Notice that det(M) is also positive.When the inclined angle  = 0, the equilibrium of the rotor angle cannot be decided based on (15); that is,  3 is an arbitrary angle.When  ∈ (0, 90], the equilibrium of  3 can be calculated as Therefore, the equilibrium of the rotor angle   can be expressed as Substituting ( 15) into ( 13), the equilibrium of the cart position   can be calculated as As a result, the equilibriums of the inclined RTAC system can be achieved by combining (18) and ( 17): Remark 4. When RTAC system is on a horizontal plane without gravity affecting it, equilibriums of the benchmark system are stable.However, the inclined RTAC system, similar to pendulum-like systems, has two types of equilibriums, that is, up equilibriums with high potential energy of gravity and down equilibriums with low potential energy of gravity.The up equilibriums are where the potential energy of gravity is −(1/ 2) 2  2 cos 2 sin 2  +  sin .And the up equilibriums are self-unstable which are not interesting in this paper.The down equilibriums are where the potential energy of gravity is −(1/ 2) 2  2 cos 2 sin 2  −  sin .In the next section, we will develop a simple linear controller for the inclined RTAC system to its typical down equilibrium from any initial conditions.
To analyze controllability around equilibriums (19), we can check if the following matrix is full rank: where Journal of Applied Mathematics 5 Calculating A, B, and C, we can get where To check the full rank condition of matrix C, one can calculate its determination as follows: Therefore, one can calculate the determination det(C  ) ̸ = 0 to find out full rank conditions of itself and C as follows: In the following, we discuss three cases to obtain the conditions on full rank of matrix C. Case 1.  = 0; that is, the RTAC system is on the horizontal plane.
Substituting the parameters , ,  into (27), determination of matrix where , ,   , det(M) are all positive parameters.Therefore, the full rank constrain for matrix C  (also matrix C) will be cos   ̸ = 0; that is, It can be observed from condition (29) that the equilibriums will be uncontrollable once the rotor angles are aligned to the moving direction of the translational cart.
Case 2. 0 <  ≤ 90 ∘ and   = (2+1.5)−,that is, the down stable equilibriums (21) of the inclined RTAC system.Calculating the determination of matrix C  similarly renders where   , , ,   , , det(M), sin  are all positive parameters.Therefore, for the down stable equilibriums of the inclined RTAC system, the controllability condition becomes According to the relationship between  and  that   = (2+1.5)−,for the down stable equilibriums of the inclined RTAC system, controllability condition (31) still implies that the equilibriums are uncontrollable when the rotor angles are aligned to the moving direction of the translational cart.
Calculating determination of matrix C  becomes as follows: Therefore, controllable conditions for the up unstable equilibriums of the inclined RTAC system on slopes are as follows: sin  ̸ = 0, (33) Comparing controllability conditions on the two cases of inclined RTAC system, they have the same controllable equilibriums.To meet the condition (34) for up unstable equilibriums, proper physical parameters of the inclined RTAC system can be considered when inclining parameters ,  are fixed.Since the up unstable equilibriums are not interesting, condition (34) will not be studied in detail in this paper.
Remark 5.It is straightforward that the actuated variable  can be controlled directly by the input torque ; however, the unactuated cart motion must be controlled through dynamical coupling of the system.The controllable equilibriums for the (horizontal or inclined) RTAC system can be explained based on (7) intuitively: the acceleration ẍ of cart's translational motion is controlled by input torque  with a coupling coefficient related to cos .If we set   to ((2 + 1)/2), rotor angle can always be brought to ((2 + 1)/2) with a simple controller.Consequently, we have cos  = 0, which means that the coupling item from input torque used to control cart motion will disappear.In other words, the input control torque  will lose control of the cart position  as soon as the rotor angle  is brought to ((2 + 1)/2).Therefore, the RTAC systems are uncontrollable when the desired rotor angles are aligned to the moving direction of the translational cart.

Energy Based Control Design
In this section, energy is employed to derive the passivity of the inclined RTAC system.Based on the passivity property, a simple PD controller is achieved for the inclined system to its controllable equilibriums from initial conditions.
Overall energy of the inclined RTAC system can be calculated with Differentiating , we can get Integrating both sides of (36), we have Therefore, based on the definition of passivity [3], the system having  as input and θ as output is passive.According to the derived passivity property of the inclined RTAC system, a control Lyapunov function candidate including system energy  is designed as where  1 > 0,  2 > 0, and are potential energy and rotor angle (here we choose  0 ∈ (−, ]) at the down stable equilibrium, respectively.Because the smallest system energy is  0 , the designed Lyapunov function ( 38) is positive definite.
Differentiating , we have According to the second Lyapunov stability theorem, the controller should guarantee the negative defined V.By choosing where  3 > 0, we get Therefore, a simple PD controller for the inclined RTAC system is derived as Finally, by using LaSalle's stability theorem [3], the global asymptotic stability of the closed-loop system consisting of dynamics (8) and controller (43) can be concluded.

Simulations
In order to verify the dynamical analysis and control design for the inclined RTAC system, simulations were programmed and performed with Matlab/Simulink.Following [5], the physical parameters of the inclined RTAC system are chosen as shown in Table 1. Figure 3 shows the simulation results to verify the system dynamics (8) and its equilibriums (19), where the left column is without friction and the right column is with friction.In the simulations, the initial conditions of the system are selected as (, ẋ , , θ ) = (0, 0, 0, 0),  = 0, and  =  = 45 ∘ which guarantees that the equilibriums are controllable according to (31) and the disturbance friction forces are modeled as   =   ẋ and   =   θ , where   ,   are the friction constants and are set as   = 0.005 and   = 0.03.For the inclined RTAC system without disturbances, whose simulation results are shown as left column of Figure 3, cart oscillates along -axis about its equilibrium (18); however, the trajectory of rotor angle  is a combination of rounds of rotation and oscillation within one round, which can be imaged because of period feature of its equilibrium (17).Once frictions are injected into the system, simulation results being shown as right column of Figure 3, both cart position and rotor angle will damp to their equilibriums: cart motion of -axis is damped oscillations with respect to its equilibrium  = −0.0383m, and the rotor angle will be stuck to  = −135 ∘ , one of its equilibriums, and damped quickly under friction.
Figure 4 shows the simulation results to illustrate the performance of the closed-loop control system consisting of system dynamics (8) and designed controller (43).Initial conditions for the simulation are (, ẋ , , θ ) = (0, 0, 0, 0),  =  = 45 ∘ , and   =   = 0. Parameters of the controller are selected as  1 = 440,  2 = 5.4, and  3 = 1.As one can see from the simulation results shown in Figure 4, the designed PD controller (43) can bring the states of inclined RTAC system from (0, 0, 0, 0) to its equilibrium (−0.0383, 0, 225, 0) in 30 s.And the stabilized rotor angle 225 ∘ here has a 360 ∘ difference compared to the rotor angle −135 ∘ as shown in Figure 3, which means the same rotor position in real physical system.From Figure 4, one also can observe that the system energy will be brought to −0.1761 J which is the lowest level.During the stabilizing process, the peak value of control torque is no more than 0.05 Nm, and the continuous control torque is no more than 0.02 Nm.
To validate the controllable condition (31) of down equilibriums of the inclined RTAC system, performance of the close-loop control system consisting of system dynamics (8) and designed controller (43) under the same initial conditions and controller parameters other than  = 0 is simulated and shown as Figure 5.The condition  = 0 which leads to the equilibrium of the rotor angle is aligned to the moving direction of the cart, which violates the controllable condition (31).One can see that the rotor angle is stabilized by the same controller in 5 s; however, the cart position is out of control and oscillates about its equilibrium −0.0542 m.Once the rotor angle is stabilized, the control output  becomes zero, and system energy is about −0.1 J which is much higher than the lowest potential energy −0.3192J.As a result, three sets of simulation results validate the dynamical analysis and control design for the inclined RTAC system in the previous sections.

Conclusion
Due to the assembly or unleveled installation base, RTAC system cannot be on an absolute horizontal plane, which is used as a benchmark for evaluating nonlinear design techniques.This paper extends the standardized RTAC system to a more general case, that is, inclined RTAC system.Detailed dynamical modeling and analysis on equilibriums show that the down stable equilibriums are controllable when the inclined direction is not aligned to the oscillating direction of the cart.And a simple PD controller was developed based on passivity of the system.Simulation results verified the correctness and feasibility of the dynamical analysis and control design for the inclined RTAC system.The dynamical analyzing result and designed controller for the inclined RTAC system can be reduced and applied to the benchmark RTAC system.

Figure 2 :
Figure 2: Mass points analysis of the inclined RTAC system.