 Research Article
 Open Access
Enhanced vibration control of a multilink flexible manipulator using filtered inverse controller
 Waweru Njeri^{1}Email authorView ORCID ID profile,
 Minoru Sasaki^{1} and
 Kojiro Matsushita^{1}
 Received: 25 June 2018
 Accepted: 9 November 2018
 Published: 26 November 2018
Abstract
Flexible manipulators have numerous advantages such as lightweight, high operation speed, and low power consumption. However, they suffer from link vibrations, especially when operated at high speeds followed by sudden stops. This limitation has been addressed using techniques such as adaptive filters, adaptive strain feedback gain, state feedback control, etc. This article presents a filtered inverse controller for the mitigation of link vibrations in a multilink flexible manipulator. To this end, the plant model, developed and linearized in Maple/Maplesim was inverted in MATLAB. The internal dynamics of the inverse model were stabilized using the state feedback technique. For safe and highspeed operations, the inverse model was augmented with a low pass filter to form the filtered inverse which was used as feedforward controller. Practical experiments were carried out in the dSPACE environment. Results show that filtered inverse controller yield not only faster response but relatively minimal link vibration when compared with the manipulator without vibration controller.
Keywords
 Inverse system
 Internal dynamics
 Link vibration
Introduction
Flexible manipulators have become popular owing to their numerous advantages in comparison to the rigid manipulators such as light weight. This called for low power rated actuators, high speed and less material requirements which render them cheap [1, 2]. Consequently, they find applications in areas like assembling of electronic hardwares, packing and palletizing, in space exploration, precision welding and painting of vehicles.
At high operation speeds and increased loading, however, due to their elastic nature, increased inertial forces leads to the vibration of the links. This causes delays in the precise positioning of the end effector. For example, approximately a third of the time spent in the operation of the remote manipulator system of the space shuttle is used to wait for vibrations to decay to negligible values [3].
One of the earliest technique of dealing with precise positioning of the end effector by damping out the link vibrations is the state feedback control. In [4], the application of state feedback on a two link, two joints system with distributed flexibility is presented. In their work, the authors derived the dynamic model of the arm using combination of transfer matrices, numerical methods and lagrangian mechanics. This was followed by the exploration of feedback schemes involving joint angles, joint velocities and flexible states. The strength of this technique lies in its simplicity. Similar work related to state feedback include [5] where LQG optimal controller is developed and employed on a model obtained by identification. Recently, authors in [6] examined accelerometer based feedback together with input shaping technique.
Yet another vibration control measure is the use of digital filters and wave shaping. Conditioning of the manipulator excitation signal can reduce the link vibration. In [7], the authors employ a lowpass digital filter to attenuate some frequency contents on the input signal thereby suppressing link vibrations. Finite impulse response (FIR) and infinite impulse response (IIR) filters with bandstop frequency response have also been designed to eliminate the dominant strain frequencies in the feedback loop. The main limitation of these fixed frequency response filters is that changes in loading and trajectories may introduce new modes different from the design frequencies or shift the frequency response of the manipulator.
The limitations of the fixed digital filters was addressed by Sasaki et al. in [8, 9]. They used adaptive notch filters and filteredX LMS algorithm based notch filter to adapt to the strain frequencies and other modes that may arise with changes in the loading. The main challenge with adaptive filters is their narrow bandwidth such that some modes will fall outside its narrow notch. These modes will thus not be adequately suppressed. The other challenge is that though the notch can shift in the spectrum, it can only do so over finite frequency range impairing the vibration suppression adaptation capability.
Researchers [10, 11] noted that vibrations are prevalent when the arms were suddenly brought to a stop. The intensity of the excited vibrations depend on the velocity prior to this sudden stop. In this regard, they proposed an input preshaping method utilizing triangular and trapezoidal velocity profile. In this scheme, the trajectory is partitioned into three sections: acceleration time, constant velocity time and deceleration time. Timings of the three sections are dependent of the final desired angle and the loading such that the arm will decelerate to a stop exciting very minimal vibration.
There are other solutions involving infinite dimensional controllers. These includes direct strain feedback controller [12], energy based robust controller [13], proportional, derivative and strain (PDS) controller [14] to mention just but a few.
Inversion of dynamics system has a history that goes back to as early as 1960s first developed by Brocket and Mesarovic in 1965 [16]. Silverman [17] in 1969 developed an iterative inversion schemes for multipleinput–multipleoutput (MIMO) for time invariant as well as time variant systems by successive differentiation and partitioning the output variable. Over and above its simplicity and efficiency, the algorithm could test for the existence of the inverse such that prior check was not necessary. Massey and Sain [18] in the same period discussed the existence, properties and the development scheme of inverses of linear systems. As a result, their quest yielded a different inversion method from that of Silverman. In the decade that followed, Moylan [15] refined the previous work on inversion and developed another algorithm with enhanced efficiency and an inverse model stability criterion. Hirschorn extended the procedures earlier developed for linear system inversion to nonlinear systems.
Key concerns in the development of an inverse model are the existence of the inverse and its stability. The aforementioned classical techniques, though very effective, were limited to minimum phase systems. For nonminimum phase systems, the yielded inverse were unstable. Devasia [19], an author who has done remarkable research in inversion theory, especially for nonminimum phase systems, successfully managed to invert a non minimum phase system by isolating the internal from the external dynamic then decomposing them into stable and unstable dynamics. He used the preview technique to solve for the unstable internal dynamics. Detailed mathematical presentation of the preview based technique can be found in [20–22]. Other inversion techniques can be found in [23, 24] and the references therein.
In this article, we developed an inverse controller and proposed a controller to stabilize the internal dynamics of the otherwise unstable inverse model. To limit operation speeds to safe levels, the stable inverse model was augmented with lowpass filters and used as a feedforward controller to a two link, 3D flexible manipulator. The main difference between the proposed approach and input shaping is that whereas the trajectories are fixed in the latter method, different joint trajectories can be used with inverse controller.
The rest of the article is organized as follows: "Model formulation and validation" section presents the modelling of the two link, 3D flexible manipulator and validation of the model against the actual manipulator. Development of the inverse model is highlighted in "Development of the inverse system" section. The manipulator model is inverted in "Inverting the multilink flexible manipulator" section. Simulation and experimental results are presented and discussed in "Results and discussion" section followed by conclusion in "Conclusion" section.
Model formulation and validation
Controller design requires the knowledge of the plant to be controlled. Consequently, accurate modelling forms a prerequisite for a controller design. A lot of research has been done in the accurate mathematical modelling of flexible manipulators, for example [25–27]. This involves the application of Lagrangian mechanics or the Euler–Newton formulation which are derived from energy principles. Since these techniques involves the solution of differential equations, the solution are truncated using either finite element method (FEM) or the assumed modes methods (AMM). Mathematical modelling other than being very tedious and prone to errors especially with growing number of links and joints, also fails to capture all the details of a plant. Alternative solutions include system identification and symbolic modelling.
In system identification, input–output data and previous knowledge of the system are used to develop a statistical or neural network model of the dynamic system whose characteristics match in one form or another the input–output relationship. The main limitation of system identification is the fact that some of the phenomenon of the original plant cannot be deduced from the input output relationship. In the same respect, behaviours outside the test data cannot be identified.
Symbolic modelling on the other hand involve the use of computer applications to model and simulate the plant. Mathematical representation of the plant is then obtained, either in state space or in differential algebraic equations (DAEs). The strength of this method lies in its accuracy owing to the fact that all technical complexities, interactions and aspects of the plant that cannot be captured mathematically are put into consideration [28, 29].
The plant presented in this article is a two link, 3D flexible manipulator with a weight attached at the distal end and structured as in Fig. 2a (see Table 2 in Appendix 2 for specifications). It has three rotary joints driven by dc servomotors and two flexible links assumed to have damping of the Kelvin–Voigt type both in the lateral and the torsional senses. The control system consists of a computer, AD and DA converters interfaced to Matlab and controlled in dSPACE control desk environment. Measurement of angular position and velocity is achieved using encoders coupled to the servomotors while link strain measurement is done by strain gauges positioned at the bottom of each link.
The manipulator was modelled and linearized in Maple/Maplesim\(^{\copyright }\). Maplesim employs the Rayleigh beam theory which incorporates the rotary inertia effect over and above the kinetic and potential energies of the bending effects considered in the popular Euler–Bernoulli beam theory [30]. Graph theory is used in the formulation of the governing dynamic algebraic equations (DAE) of the flexible manipulator which will take the form of a system of PDEs and boundary conditions in form of ODEs. Simulations with such infinite dimensional PDEs would require impractical resources in terms of computer memory and would take a long time. To solve this, the truncation of the elastic coordinates for the deformation along each axis amongst the inplane, outofplane, torsional deflections and longitudinal elongation is done using the assumed mode method. In this work, the manipulator was truncated to order 2.

\(x_{1}(t) = i_1(t)\) \(\bullet\) \(x_{7}(t) = \dot{w}_{21}(t)\) \(\bullet\) \(x_{13}(t) = \dot{\theta }_2(t)\)

\(x_{2}(t) = w_{11}(t)\) \(\bullet\) \(x_{8}(t) = w_{22}(t)\) \(\bullet\) \(x_{14}(t) = \theta _3(t)\)

\(x_{3}(t) = \dot{w}_{11}(t)\) \(\bullet\) \(x_{9}(t) = \dot{w}_{22}(t)\) \(\bullet\) \(x_{15}(t) = \dot{\theta }_3(t)\)

\(x_{4}(t) = w_{12}(t)\) \(\bullet\) \(x_{10}(t) = \theta _1(t)\) \(\bullet\) \(x_{16}(t) = i_3(t)\)

\(x_{5}(t) = \dot{w}_{12}(t)\) \(\bullet\) \(x_{11}(t) = \dot{\theta }_1(t)\) \(\bullet\) \(x_{17}(t) = i_2(t)\)

\(x_{6}(t) = w_{21}(t)\) \(\bullet\) \(x_{12}(t) = \theta _2(t)\)
Remark 1
In the modelling of the manipulator in maplesim, the lengths of links 1 and 2 are broken into two to accommodate an instrument to measure the strain. In regard to this, in the linearized model, the flexure variable has two parts as \(w_{11}\), \(w_{12}\) for link 1 and \(w_{21}\), \(w_{22}\) for link 2, where the variable \(w_{ij}\) represent deflection in the jth (j = 1, 2) part of link i(\(i = 1, 2\)) respectively. Except for having twice as many flexure variables as the number of links, breaking the links does not affect the performance of the model.
To validate the Maplesim model, Figs. 3 and 4 shows the joint angles and strain information of the nonlinear model and the linear model against the actual manipulator. We can see perfect agreement between joint angles in 3a–3c, torsional and links strain in 4a–4c. From this observation, we can deduce that the linear model represented an accurate model of the manipulator. Further, inverse model developed from this model was an accurate inverse of the model and that of the actual manipulator.
Development of the inverse system
Definition 1
Given \(\sum (t)\), an LTI system defined above in Eqs. (1) and (2), inversion involves the development of a model \(\sum ^{1}(t)\) that yields the input control law \(u_f(t)\) to reproduces y(t) when used as the input to \(\sum (t)\).
Definition 2
If \(C_i\) denotes the \(i_{th}\) row of the output matrix C, then the system is said to have a relative degree \(r \triangleq (r_1,r_2\ldots ,r_p)^T\) if \(C_iA^lB = 0\), \(\forall l < r_i1\); \(1\le i\le p\) [31]. Further, if this holds true in the entire domain in the states, then we say the system has a well defined relative degree.
Inverting the multilink flexible manipulator
Results and discussion
Whereas the filter is controlling the speed of response and thereby suppressing the vibrations, the role of the inverse controller is in ensuring that the joints follows the desired trajectory. The filter on its own will result in a system commanded by a signal whose high frequency components have been filtered out and will hardly be able to track the desired joint angles as is seen in Figs. 9 and 10.
Conclusion
In this paper, we successfully developed a model of a two link, 3D flexible manipulator, linearized it before developing an inverse model. The inverse model was augmented with order \(n = 2\) filter and used as a feedforward controller. The filter and the inverse model were integrated in an already existing Matlab model and the experiments carried out in dSPACE control environment. Comparing with a system without the inverse controller, the results presented show a significant reduction of the link vibration while closely tracking the desired joint angles as facilitated by the inverse controller. We found that the links velocities were dependent on the value of the filter time constant \(\lambda\) whose careful choice could yield high operation speeds and minimal link vibrations. The contribution of this article is the development of a filter augmented inverse controller and the application of this controller in vibration control akin to input shaping technique. The merit of this technique is that the trajectory is not fixed like in classical input shaping methods. Except for disturbance handling capabilities, the filtered inverse controller is simple to implement and accurate without the known limitations affecting other methods like PID, i.e. integral windup and high frequency noise amplification. To improve on the disturbance handling capabilities of the filtered inverse controller, this work can be extended to internal model control (IMC) system.
Declarations
Authors' contributions
WN, MS and KM contributed to the concepts and the experimental design, WN carried out the experiments of this study, did the analysis and drafted the manuscript. MS and KM revised and refined the manuscript. All authors read and approved the final manuscript.
Acknowlegements
Not applicable.
Funding
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Availability of data and materials
Not applicable.
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