Input-Shaping for Feed-Forward Control of Cable-Driven Parallel Robots
Sana Baklouti, Eric Courteille, Philippe Lemoine, Centrale Nantes, Stéphane Caro
IInput-Shaping for Feed-Forward Control ofCable-Driven Parallel Robots
Sana Baklouti
University of Nantes,
IUT Nantes, LS2N UMR CNRS 6004,2 Avenue du Professeur Jean Rouxel,44475 Carquefou, FranceEmail: [email protected]
Eric Courteille
University of Rennes,
INSA Rennes, LGCGM-EA 3913,20, avenue des Buttes de C¨oesmes,35043 Rennes Cedex, FranceEmail: [email protected]
Philippe Lemoine
Centrale Nantes,
LS2N UMR CNRS 6004,1, rue de la No¨e,44321 Nantes Cedex 03, FranceEmail: [email protected]
St ´ephane Caro ∗ CNRS,
LS2N UMR CNRS 6004,1, rue de la No¨e,44321 Nantes Cedex 03, FranceEmail: [email protected]
ABSTRACT
This paper deals with the use of input-shaping filters in conjunction with a feed-forward control of Cable-Driven Parallel Robots (CDPRs), while integrating cable tension calculation to satisfy positive cable tensionsalong the prescribed trajectory of the moving-platform. This method aims to attenuate the oscillatorymotions of the moving-platform. Thus, the input signal is modified to make it self-cancel residual vibrations.The effectiveness, in terms of moving-platform oscillation attenuation, of the proposed closed-loop controlmethod combined with shaping inputs is experimentally studied on a suspended and non-redundant CDPRprototype. This confirms residual vibration reduction improvement with respect to the unshaped control interms of Peak-to-Peak amplitude of velocity error, which can achieve 72 % while using input-shaping filters.
Cable-Driven Parallel Robots (CDPRs) are a particular class of parallel robots, where the rigid links arereplaced by cables. They consist of a moving-platform connected to a base with cables. Thanks to their large ∗ Corresponding author a r X i v : . [ c s . R O ] S e p ayload capacity, their high dynamic performance and large workspace with respect to their dimensions, CDPRs canbe used for several types of industrial applications such as additive manufacturing [1], painting, sand blasting [2] andassembly [3]. As safety fulfillments are taken into consideration for most of CDPR applications, these manipulatorscan also be used in search and rescue [4] and rehabilitation [5] operations.CDPRs can reach high velocities and accelerations in large workspaces thanks to their low inertia [6]. However,vibrations may occur. Pose stabilization and/or trajectory tracking of the moving-platform can be degraded due tocable elasticity. Considering the physical cable characteristics, the cable elasticity has mainly two origins. Thefirst one is the axial stiffness of the cables, which is associated with the elastic material modulus and the cablestructure. The second is the sag-introduced flexibility, which comes from the effect of cable weight onto the staticcable profile [7, 8].CDPR accuracy improvement is still possible once the robot is manipulated through a suitable control scheme.Several controllers have been proposed in the literature to improve CDPR accuracy locally or along a giventrajectory [9, 10, 11, 12]. Some papers deal with the CDPR control while considering cable elongations and theireffect on the dynamic behavior. In [13], an approach of wave based control (WBC) is proposed for large scalerobots whose cables sagging effect cannot be neglected. It combines the position control and the active vibrationdamping simultaneously. This control strategy assumes actuator motion as launching a mechanical wave into theflexible system, which is absorbed on its return to the actuator. The assumption of modeling cables as elasticstraight and massless links is valid for robots with relatively small size [14]. The cable mass is supposed to benegligible with respect to the moving-platform mass [15,16]. The dynamic modeling and adaptive control of a singledegree-of-freedom flexible cable-driven parallel robot (CDPR) is investigated in [17]. A Rayleigh-Ritz cable modelis developed that takes into account the changes in cable mass and stiffness due to its winding and unwindingaround the actuating winch, with the changes distributed throughout the cables. A robust H ∞ control scheme forCDPR is described in [18] while considering the cable elongations into the dynamic model of the moving-platformand cable tension limits. Besides, H ∞ control scheme for position control of 6-DOF CDPRs is proposed in [19].Compared to [18], the position control scheme is done in the operational space and the tension management ismade separately in a more efficient way. The control of CDPRs while assuming flexibility in cables is proposedin [20, 21, 22]. In [23, 24, 25], the undesirable vibrations are attenuated by using the singular perturbation theory,which is based on measured cable elongations. Usually, cables are modeled by linear axial springs, with a constantstiffness. In [26], variable cable stiffness, which is a function of cable length variation is considered into the controlloop. In this context, authors of [27] have also proposed a robust adaptive controller to attenuate vibrations inpresence of kinematic and dynamic uncertainties. This control method requires the measurement of cable lengthsand moving-platform pose. However, the use of exteroceptive sensors to measure the moving-platform pose increasesthe complexity of the overall system [28].The importance of the feed-forward effect on non-linear systems control is highlighted in [29]. It leads to stablesystems with enhanced trajectory tracking performances. Feed-forward model-based controllers are used to fulfillaccuracy improvement by using a CDPR reference model [30]. This latter predicts the mechanical behavior of therobot; and then generates an adequate reference signal to be followed by the CDPR. This type of control providesthe compensation of the desirable effects without exteroceptive measurements. A model-based control schemefor CDPR used as a high rack storage is presented in [31]. This research work takes into account the mechanicalproperties of cables, namely their elasticity. This strategy, integrating the mechanical behavior of cables in thereference signal, enhances the CDPR performance. Although cable stiffness is considered, a limitation of thiscontrol method lies in the cable interactions with the overall system that are not considered, which can result inunwanted oscillations of the moving-platform. A novel model-based control strategy allowing the pre-compensationnot only of cable elongations but also the effect of their interaction with the overall system was proposed in [32, 33].This control method reduces the oscillations of the moving-platform.To reduce the moving-platform vibrations, a frequency dependent method, named as input-shaping [34], wasproposed for the closed-loop control of flexible systems. Input-shaping method consists in re-designing the desiredcommand signal such that the robot self-cancels residual vibrations. It is used for manipulators with flexible jointsto remedy the resulting oscillations. Input shaping was used for the control of serial robots [35, 36, 37] such as theindustrial SCARA manipulator [38]. Oscillation control by shaping the input signal was also applied for conventionalparallel robots [39, 40, 41]. Input-shaping was also used for the control of humanoid robots [42, 43, 44] to deal withunwanted vibrations resulting from their non-rigid behavior. Once control methods are not allowing all DOF of themoving-platform of an under-actuated CDPR to be controlled, the extra DOFs in motion of the moving-platformmakes it easy to sway [45]. This leads to low operational efficiency and then the lost of controllability. Input-shapingmethod for under-actuated CDPRs was proposed as an alternative to attenuate residual vibrations [46, 47, 48, 49, 50].Input-shaping was also proposed in [51] for over-actuated CDPRs. However, this control does not allow to managethe actuation redundancy. CDPR vibration reduction approach using input-shapers is presented in [52]. Thiscontrol method integrates a feedback controller, whose linearization is based on CDPR rigid model.2ccordingly, this paper deals with the input-shaping for feed-forward control of CDPRs as a mean to attenuatethe oscillations of their moving-platform. The main contribution of this paper lies in the use of input-shaping filtersin conjunction with an experimentally validated model-based feed-forward control [33] for disturbance rejection,while considering the overall stiffness of the system. This control method integrates cable tension calculation tosatisfy positive cable tensions along the prescribed trajectory of the moving-platform.This paper is organized as follows: Section 2 describes the CDPR parameterization and gives the equationsrequired to establish the control law. Section 3.1 introduces the principle and the design of input-shaping filters.Then, the application of input-shaping for feed-forward closed-loop control is presented in Section 3.2. Experimentalvalidations performed on the CREATOR prototype located at LS2N, Nantes, France, a CDPR with three cablesand three Degree-Of-Freedom (DOF), are discussed in Section 4. The setup of the input-shapers is based on arobustness analysis as presented in Section 4.2. Finally, the effectiveness, in terms of moving-platform oscillationattenuation, of the proposed closed-loop control method combined with shaping inputs is experimentally studied inSection 4.3. A CDPR with n cables, whose geometry is illustrated in Fig. 1, has n anchor points, denoted by A i and n exitpoints, denoted by B i , i ∈ [1 n ]. The Cartesian coordinate vectors of anchor points A i and exit points B i of aCDPR, i ∈ [1 n ], are denoted a i and b i , respectively. These vectors are expressed in the moving-platform frame F p = { P, x p , y p , z p } and in the base frame F b = { O, x b , y b , z b } , respectively.Fig. 1: The i th closed-loop of a CDPRThe pose x = [ p T o T ] T ∈ R m of the moving-platform geometric center P in the base frame F b is describedby the position vector p = [ x, y, z ] T ∈ R u and the orientation vector o = [ φ, θ, ψ ] T ∈ R v . The orientation ofthe moving-platform is parameterized by Euler angles φ , θ and ψ . u being the number of translational Degrees-Of-Freedom (DOFs), v being the number of rotational DOFs and m being the total DOFs of the moving-platform.The geometric closed-loop equations for a CDPR with straight line cables take the form: l i = b i − a i − p , i = [1 n ] , (1)3here l = [ l , ..., l n ] T ∈ R n is the cable length vector and l i = k l i k is the i th cable length. The inverse kinematicsof the CDPR is expressed as follows: ˙l = At , A = u a × u . .. . u n a n × u n , (2)where A ∈ R n × m is the Jacobian matrix, which is a function of x and maps the moving-platform velocities tothe cable velocity vector. u i ∈ R u is unit vector of i th cable pointing from A i to B i . t = [ ˙p T ω T ] ∈ R m is themoving-platform twist and ω = [ ω x , ω y , ω z ] T is its angular velocity vector.The actuator angular displacement and velocity vectors are denoted as q = [ q , ..., q n ] T ∈ R n and˙ q = [ ˙ q , ..., ˙ q n ] T ∈ R n , respectively. The relationship between q ( ˙q , resp.) and l ( ˙l , resp.) is linear: q = χ − l , ˙q = χ − ˙l , (3)where χ = diag [ χ , ..., χ n ] ∈ R n × n is a diagonal matrix presenting the winches winding ratio.The equations of motions are derived from the equations of Newton-Euler: M ˙t + C t = W τ + w ex , (4)with M = (cid:20) m ee I m − m ee d × m ee d × I ee − m ee d × d × (cid:21) , C t = (cid:20) m ee ω × ω × d ω × (cid:0) I ee − m ee d × d × ω (cid:1)(cid:21) , W = − A T . (5) ˙t = [ ¨p T ˙r T ] ∈ R m is the acceleration vector of the moving-platform. m ee is the total mass of the moving-platform. d = −−→ P G = [ d x , d y , d z ] T is the vector pointing from the moving-platform geometric center P to mass center G . M ∈ R m × m is the generalized mass matrix of the moving platform . I ee denotes the inertia matrix of themoving-platform expressed at its center of mass. C ∈ R m × m is the matrix of Coriolis and centrifugal forces. w ex ∈ R m is the external wrench applied onto the moving-platform. τ = [ τ , .., τ n ] T ∈ R n is the cable tensionvector, τ i being the tension in cable C i , i = [1 ..n ].The calculation of cable tension vector τ presents two cases as a function of the degree of the CDPR actuationredundancy. The first case is when the number of cables n is equal to the DOF m . In this case, the determinationof cable tensions leads to only one solution as long as the wrench matrix W is not singular. The second case ariseswhen m < n . The redundancy allows to select a solution amongst the infinite set cable tension vectors as long asthe moving-platform pose is wrench feasible . For many CDPRs, the axial stiffness of the cable, which is a function of its elastic modulus, is the main sourceof flexibility, which must be considered into the CDPR model [54]. Admitting cable tension model (either linear, i.e. directly proportional to cable elongation [32], or non-linear, i.e. a logarithmic function of cable elongation [14]), thestiffness matrix is expressed as a function of the moving-platform pose, the CDPR geometry and the cable tensions.Assuming all cables are sufficiently tensioned, and submitted to low strains and small elongations, we canconsider the linear cable tension model: τ = K l δ l . (6) ω × = " − ω z ω y ω z − ω x − ω y ω x , d × = " − d z d y d z − d x − d y d x . A cable force distribution is said to be feasible in a particular configuration and for a specified set of wrenches, if the tension forcesin the cables can counteract any external wrench of the specified set applied to the moving-platform [53]. l = diag [ k , ..., k n ] ∈ R n × n being a diagonal matrix presenting the i th cable axial stiffness k i = EAl i . δ l = [ δl , ..., δl n ] T ∈ R n is the cable elongation vector. E is the cable modulus of elasticity and A is the cross-sectionof the cable.As δ l = A δ x , the generalized overall stiffness matrix K x of the CDPR is expressed as follows: K x = A T K l A . (7) δ x being the moving-platform displacement screw corresponding to the cable elongation vector δ l from their staticequilibrium configuration. Therefore, the moving-platform free vibration around any equilibrium is managed bythe following linear perturbation equation: M δ ˙t + C δ t + K x δ x = . (8) As the input-shaping method is frequency dependent, the knowledge of natural frequencies of the CDPR isessential. The natural frequencies of the CDPR can be calculated by solving the classic eigenvalue problem basedon the global stiffness matrix of the CDPR as mentioned in [55]. They are obtained by solving the generalizedeigenvalue problem associated with the generalized mass matrix M and stiffness matrix K x , which is described asfollows: det( λ I m − M − K x ) = 0 , (9)where I m ∈ R m × m is the identity matrix. The i th natural frequency f i (Hz) corresponds to the i th solution λ i ofthe characteristic polynomial described by Eq. (9), f i = √ λ i π .Note that the natural frequencies can be calculated considering the cable damping through the use of a complexYoung modulus. In that case, Dynamic Mechanical analysis (DMA) can be used to identify the cable stiffness andcable damping under forced oscillatory measurements. In [14], it was shown that the elastic modulus and dampingthus obtained are highly dependent on frequency for a given preload, over a representative frequency range of theCDPR behavior. This section deals with the principle and the design of input-shaping filters leading to the reduction of residualvibrations by generating a self-canceling control signal. The application of these filters into the closed-loopfeed-forward control of CDPRs is then introduced.
Input shaping principle consists of a convolution of the control inputs with series of impulses, each described byan impulse amplitude and a delay [56]. Those coefficients are chosen in a way that the sum of residual vibrationsproduced by each impulse cancel each other and produces a reference trajectory slightly different from the originalone, which does not produce residual vibrations of the system.Figure 2 shows the considered control architecture. The input shaper filters modify the desired moving-platformtrajectory (pose x d , velocity t d and acceleration ˙t d ) and provides a shaped trajectory (pose ˆ x d , velocity ˆ t d andacceleration ˆ ˙ t d ), which becomes the desired motion of the moving-platform. If the input shaper is properly designed,then the new shaped command will lead to a response for which the residual vibration is reduced or eliminated asexplained in [47].Two types of input-shapers are distinguished: single-mode and multi-mode input-shapers. The behavior ofa k -mode input shaper can be described by the applied sequence of impulses S defined by the following transfer5ig. 2: Block diagram: Input-shaping for closed-loop control schemefunction: S ( p ) = ˆ x d ( p ) x d ( p ) = ˆ t d ( p ) t d ( p ) = ˆ˙ t d ( p )˙ t d ( p ) = k X i =1 D i e − pt i , (10)where k is the number of considered vibration modes into input-shaping filters design. p is the Laplace variable. D i is the i th mode impulse amplitude vector and is a function of the natural frequency f i . Its corresponding dampingratio is ζ i . t i is the time location of the applied impulse.Zero-Vibration (ZV) shaper is a single mode input-shaper. It amounts to the convolution of the original inputsignal with a sequence of two impulses. The impulses are separated by the half of the robot’s natural period. TheZV-shaper for i th natural mode is expressed as follows: ZV i = (cid:20) D i t i (cid:21) =
11 + (cid:15) i (cid:15) i (cid:15) i f i , (cid:15) i = e − ζ i π q − ζ i . (11)Zero-Vibration-Derivative (ZVD) is a single mode input-shaper. It consists in the convolution of the originalinput signal with a sequence of three impulses separated by half the period of the robot’s vibration. The ZVD-shaperfor i th natural mode is expressed as follows: ZVD i = (cid:20) D i t i (cid:21) =
11 + 2 (cid:15) i + (cid:15) i (cid:15) i (cid:15) i + (cid:15) i (cid:15) i (cid:15) i + (cid:15) i f i f i . (12)A convolution of multiple single mode shapers aims to control multiple modes of a robot manipulator [57]. Asimple way to obtain a two-mode shaper is to convolve two single-mode shapers [58]. In fact, the ZV-ZV shaper(ZVD-ZVD shaper, resp.) consists of the convolution of two ZV shapers (ZVD shapers, resp.) applied for i th and j th natural modes: ZV − ZV i − j = (cid:20) D i − j t i − j (cid:21) = (cid:15) i + 1)( (cid:15) j + 1) (cid:15) j ( (cid:15) i + 1)( (cid:15) j + 1) (cid:15) i ( (cid:15) i + 1)( (cid:15) j + 1) (cid:15) i (cid:15) j ( (cid:15) i + 1)( (cid:15) j + 1)0 12 f j f i f i + f j ) (13)6nd ZVD − ZVD i − j = (cid:20) D i − j t i − j (cid:21) , D i − j = 1( (cid:15) i + 1) ( (cid:15) j + 1) (cid:2) (cid:15) j (cid:15) j (cid:15) i (cid:15) i (cid:15) j (cid:15) i (cid:15) j (cid:15) i (cid:15) i (cid:15) j (cid:15) i (cid:15) j (cid:3) , t i − j = (cid:20) f j f j f i f i + f j f i f j f i + f j f i f j f i f j + f i f j f i f i + f j f i f j (cid:21) (14)The integration of an input-shaping filter leads to oscillation attenuation, but not to cancellation even if amulti-mode shaper is used. This fact can be related to uncertainties in the CDPR model leading to frequencyidentification errors. Figure 3 presents the application of input-shaping into closed-loop feed-forward control, based on a classicalPID control. The shaped motion of the moving-platform (pose ˆ x d , velocity ˆ t d and acceleration ˆ˙ t d ) becomes thedesired one for the moving-platform. The vectors q d , ˙q d and ¨q d present the winch angular displacement vector,the winch velocity vector and the winch acceleration vector, respectively, corresponding to the shaped motion ofthe moving-platform (pose ˆ x d , velocity ˆ t d and acceleration ˆ˙ t d ).Fig. 3: Feed-forward closed-loop control schemeThe torque set-point Γ assigned to the motors is expressed as: Γ = Γ corr + Γ f ( ˙q d ) + Γ d , (15)where Γ d = χ − τ d ∈ R n is the torque set-point corresponding to the tension calculated along the shaped trajectory. w f ∈ R m denotes the external wrench and is the input signal of the tension calculation block. It is calculated basedon the generated trajectory and the external wrench applied on the moving-platform, w f = M ˆ˙ t d + C ˆ t d − w ex . It isabout solving the dynamic equilibrium equations for a given pose x of the moving-platform, which can be describedas follows: W τ d + w f = , (16)The cables can only pull and not push the moving-platform. If the number of cables n is equal to the DOF m , the7nversion of Eq. (16) is possible, as long as the wrench matrix W is not singular. However, this tension set canpresent negative values. That means that one or more cable(s) may have to push the moving-platform, which isnot possible. When m < n , Eq. (16) may have an infinite number of solutions. Therefore, the redundancy allowsto select a solution amongst the infinite set cable tension vectors satisfying some criteria. The problem of forcedistribution presents one important control issue for redundant actuated CDPRs, which is the determination offeasible cable force distribution [59, 60, 61, 62]. Γ f ( ˙q d ) is the joint friction torque vector, which is expressed according to the static model of friction [63] asfollows: Γ f ( ˙q d ) = Γ s sgn( ˙q d ) + Γ v ˙q d , (17)where Γ s ∈ R n × n is a diagonal matrix containing the dry friction coefficients and Γ v ∈ R n × n is a diagonal matrixcontaining the viscous friction coefficients. Γ corr = I m h ( t ) corresponds to the torque of correction, where I m is a diagonal matrix containing the winchmoment of inertia. h ( t ) is defined by: h ( t ) = ¨q d + K p ( q d − q ) + K d ( ˙q d − ˙q ) + K i Z t + t ( q d − q ) dt, (18)where K p ∈ R n × n is the proportional gain matrix, K d ∈ R n × n is the derivative gain matrix, K i ∈ R n × n is theintegrator gain matrix. q is the measured angular displacement vector of motors. The closed-loop dynamicscorresponds to the following tracking error equation: ¨e + K p e + K d ˙e + K i Z t + t e dt = , (19)where e = q d − q and ˙e = ˙q d − ˙q are the tracking errors. Experimentations have been performed here to analyze the effect of the integration of input-shaping filters intothe closed-loop control scheme. A suspended configuration of the reconfigurable CREATOR prototype, shownin Fig. 4, with 3 cables and 3 DOF is studied. The size of this prototype is about 4.5 m long, 4 m wide and3 m high. The Cartesian coordinates of exit points B i expressed in F b are: b = [ − . , . , . T m , b = [2 . , . , . T m and b = [ − . , − . , . T m . Those coordinates were measured with aRADIAN Laser tracker . The nominal mass of the moving-platform is equal to 0.650 kg. It is supposed to be apoint-mass. The reader is referred to [33] for more details about the identification procedure.The CREATOR cables are made up of eight threads of polyethylene fiber with a diameter of 0.5 mm. Thesecables were experimentally upstream identified. The identification method is described in [33]. The absoluteuncertainties in the applied force and resulting elongation measurements from the test bench outputs are estimatedto be ± ± ± ™ motors with gearboxes, whose moment of inertia I m is equal to 0.0031 kg.m . This set is connected to 3D printed winches. Each motor is connected to a Parker ™ motor drive, which communicates with the dSpace ™ controller through bi-directional real-time links.The friction torques of the actuators are identified with respect to the static friction model [63] by incrementingthe joint angular velocity. As we do not have an accurate measurement of the motor torques, we suppose that theviscous friction torque is null. Here, the dry friction value is equal to 0.14 N.m.The equivalent architecture of the CREATOR prototype is described in Fig. 5. The command of the CREATORprototype is implemented in a host PC through a software interface generated by ControlDesk ® . This latter A cable force distribution will be said to be feasible in a particular configuration and for a specified set of wrenches if the cabletensions can counteract any external wrench of the specified set applied to the moving-platform [53]. The static measurement accuracy of the used laser tracker is ± µ m. Its range of working is between 0 and 25 m. ControlDesk is the dSPACE experiment software for seamless electronic control unit development. It performs all the necessarytasks and gives a single working environment, from the start of experimentation right to the end. ® , in the dSpace ™ controlunit. Fig. 5: Main hardware components of the CREATOR protypeThe used gains K p , K d and K i of the PID controller are equal to 1125.8 N.m, 58.12 N.m and 7269.60 N.m,respectively. A non-smooth velocity trajectory is chosen to excite the natural modes of the CDPR. The motion is uniformlyaccelerated until the moving-platform achieves a desired position along x -axis, y -axis and z -axis. The accelerationsof the moving-platform along x -axis, y -axis and z -axis are defined by a bang-bang profile (Fig. 6c).The nominal trajectory of the moving-platform is a vertical straight line (Fig 6a) from point P of Cartesian coor-dinate vector p = [0 . , − . , . T m to point P of Cartesian coordinate vector p = [0 . , − . , . T m9 Time (s) -0.500.511.52 ( m ) P o s iti on xyz (a) Time (s) ( m / s ) ˙ x ˙ y ˙ z (b) Time (s) -0.500.5 A cce l e r a ti on ( m / s ) ¨ x ¨ y ¨ z (c) Fig. 6: Nominal (a) position (b) velocity (c) acceleration profiles of the moving-platformduring t f = 3 s. The nominal velocity (Fig. 6b) and acceleration (Fig. 6c) profiles of the moving-platform arethe time derivatives of the nominal trajectory. Therefore, the trajectory followed by the moving-platform isparametrized as follows: p ( t ) = β t + β t + β , t ∈ [0 t f , (20a) p ( t ) = β ( t − t f ) + β ( t − t f ) + β , t ∈ [ t f t f ] , (20b) p ( t ) = p , t > t f , (20c)where β = β = 2 ( p − p ) t f , β = p , β = p , β = β = . (21) The first calculated natural frequencies of the CREATOR prototype under study are f = 3.67 Hz, f = 6.34 Hzand f = 7.82 Hz, which correspond to impulse times 0.27 s, 0.15 s and 0.12 s. Those natural frequencies are10alculated at the initial pose of the moving-platform trajectory by solving the generalized eigenvalue problemassociated with the apparent stiffness of the CDPR as mentioned in Section 2. Note that the natural frequenciesare calculated while neglecting cable damping. The residual vibrations are cancelled when the shaper parametersare perfectly tuned; conversely an error on the identification of the mode frequencies induces residual vibrations.Therefore, the shapers setup is preceded by a robustness analysis to errors in frequency identification.Figure 7 displays the sensitivity curves for single mode ZV and ZVD shapers [64]. It shows the amplitudeof residual vibrations as a function of the normalized frequency f /f m . f m is the predicted value of the naturalfrequency obtained from the model and f is the effective natural frequency of the CDPR for a given end-effectorpose. Here, the frequency f m used to tune the input-shaper is set to 3 .
67 Hz and the corresponding damping ratio ζ to zero. The vibration percentage is the ratio in percentage between the amplitude of vibrations when inputshaping is used and the amplitude of residual vibrations when shaping is not used.Fig. 7: Sensitivity curves of ZV and ZVD input-shapersThe sensitivity curve of the ZV-shaper shows that the larger the modeling errors, the faster the residualvibration increases when using a ZV-shaper. Besides, the vibrations obtained when using a ZVD shaper remainsmall. The robustness can be measured quantitatively by measuring the width of the curve at some low levelof vibration percentage. This non-dimensional robustness measure is called the shaper insensitivity. The shaperinsensitivity is denoted as I ZV for the ZV-shaper and as I ZV D for the ZVD-shaper.The 5 % shaper insensitivity is shown in Fig. 7 and is defined as a safety limit. For the ZV-shaper, I ZV = 0 . I ZV D = 0 .
28. FromFig. 7, it is noteworthy that the ZVD-shaper is significantly more robust than the ZV-shaper to modeling errorsand to variations in the first natural frequency along the path followed by the robot end-effector.The effective natural frequency of the CDPR can be measured by using a modal test with an impact hammer ordynamic shaker around an equilibrium position or by Fourier transform of the CDPR response during an excitingtrajectory tracking. From [33], the experimental value of the first natural frequency of CREATOR is equal to3.6 Hz at the starting point of the test path, which is a value very close to the 3.67 Hz predicted by the model.Figure 8 shows the calculated values of the first natural frequency f of CREATOR prototype through theCartesian space and the test path (in magenta) followed by its end-effector. A schematic of the CDPR with itseffector in the middle of the test path is also depicted in Fig. 8. For better clarity, the contours of f are presentedalong slices spaced 0.5 m apart along the z -axis.Figure 9 shows the calculated values of f as a function of the z -coordinate, z P , of the point-mass end-effector P along the test path. The shaper insensitivities I ZV and I ZV D are represented in Fig. 9 too. Input-shapers areconsidered robust to variations on the first natural frequency when the normalized frequency does not exceed the5 % insensitivity. Here, the normalized frequency f /f m should remain between 0.97 (0.86, resp.) and 1.03 (1.14,resp.) while using a ZV shaper (ZVD shaper, resp.). Considering the value of the first natural frequency set to f m = 3.67 Hz, the variation range to guarantee the 5 % insensitivity is 0.22 Hz (from 3.57 Hz to 3.79 Hz) while usinga single mode ZV-shaper. This same variation range increases to 1.03 Hz (from 3.17 Hz to 4.21 Hz) with a single11ig. 8: Calculated values of the first natural frequency of CREATOR through the Cartesian space and the testpath (in magenta) followed by the end-effectorFig. 9: Calculated values of the first three natural frequencies of CREATOR along the test path and the bounds onthe first natural frequency for the vibration percentage to remain smaller than 5%mode ZVD-shaper. As one can see in Fig. 9, the 5 % insensitivity is practically respected over the sub-workspacefor the ZVD-shaper, while the criterion is respected only for the beginning and the end of the trajectory for theZV-shaper.The ZV-shaper sensivity workspace W ZV associated to the ZV-shaper and the calculated value f m of the first12 a) (b) Fig. 10: The ZV-shaper sensivity workspace W ZV for CREATOR and the test path shown in magenta: (a) 3Dview; (b) Sectional view for z P = 1 m (a) (b) Fig. 11: The ZVD-shaper sensivity workspace W ZV D for CREATOR and the test path shown in magenta: (a) 3Dview; (b) Sectional view for z P = 1 mnatural frequency at the starting point of the test path is shown in Fig. 10a and defined as: W ZV = { p ∈ R : 1 − I ZV ≤ f f m ≤ I ZV } (22)with p denoting the Cartesian coordinate vector of the point-mass end-effector.The ZVD-shaper sensivity workspace W ZV D associated to the ZVD-shaper and the calculated value f m ofthe first natural frequency at the starting point of the test path is shown in Fig. 11a and defined as W ZV D = { p ∈ R : 1 − I ZV D ≤ f f m ≤ I ZV D } (23)The index ν characterizing the robustness of the shaper to variations in f at a specific end-effector pose is13efined as follows: ν = min { f − (1 − I j f m , (1 + I j f m − f } , j = { ZV, ZV D } (24)Figure 10b (11b, resp.) represents the contours of ν through the sectional view of W ZV ( W ZV D , resp.) at the z -coordinate z P = 1 m. ν = 0 Hz means that the end-effector is located on the boundary of the shaper sensivityworkspace. The larger ν , the lower the vibration percentage of the end-effector (Fig 7). Figures. 10a-b and 11a-bclearly show that the area where the path can be defined from the considered starting point is much larger withthe ZVD-shaper than with ZV-shaper. It should be noted that an adaptive shaper [65, 66, 67] should be used oncethe path goes beyong the shaper sensitivity workspace. Figures 12a shows the velocity error δ ˙ z of the moving-platform along z-axis, which is defined as the differencebetween the nominal velocity of the moving-platform and the measured one. The black (red, blue, green, cyan, Time (s) -0.2-0.15-0.1-0.0500.050.10.150.2 ˙ z ( m / s ) (a) Time (s) -0.0100.010.020.030.040.050.060.07 ˙ z ( m / s ) without ISZVZV-ZVZVDZVD-ZVDMean (b) Time (s) -0.0100.010.020.030.04 ˙ z ( m / s ) (c) Fig. 12: Experimental results: Moving-platform (a) Velocity error along z-axis with and without input-shaping(b) During the steady-state phase (c) Zoom on the moving-platform velocity error with and without input-shapingduring the steady-state phaseresp.) plot depicts δ ˙ z when the unshaped (ZV-shaped, ZVD-shaped, ZV-ZV-shaped, ZVD-ZVD-shaped, resp.)motion is used as a reference. 14o compare the results obtained with the input-shaping filters, we focus on the oscillations generated bydiscontinuities. Accordingly, a zoom is made at time range t ∈ [3 6] s, as the moving-platform is supposedto become motionless at time t = 3 s and then residual vibrations appear. From Fig. 12c, it appears that theZVD-ZVD shaper is the fastest one in terms of residual vibration attenuation.To better compare the performances of the different input-shapers, Fig. 13 represents a bar chart showing thefirst Peak-to-Peak amplitude of δ ˙ z . This period starts when the δ ˙ z curve intersects with the line presenting themean value of oscillations. When no input-shaper is used, the Peak-to-Peak amplitude of δ ˙ z is equal to 0.0097 m/s. without IS ZV ZV-ZV ZVD ZVD-ZVD00.0020.0040.0060.0080.01 P ea k - t o - p ea k a m p lit ud e ( m / s ) Fig. 13: Bar chart of the first period Peak-to-Peak amplitude of δ ˙ z It is equal to 0.0061 m/s, 0.0045 m/s, 0.0056 m/s and to 0.0028 m/s when the ZV, ZVD, ZV-ZV and the ZVD-ZVDshapers are used, respectively. These results confirm that the ZVD and the ZVD-ZVD shapers lead to the moststable behavior of the moving-platform.Contrary to the ZV-shaper, the ZVD-shaper is more robust to modeling errors. The convolution of twoone-mode robust shapers results in a two-mode robust shaper dealing with two natural modes of the robot understudy. That explains the fact that the ZVD-ZVD filter leads to better oscillation attenuations. The two-modeZVD-ZVD input-shaping filter leads to the most robust control scheme and to the best one in terms of vibrationrejection. Nevertheless, this robustness incurs in a time penalty (a delay of 0.23 s with ZVD-ZVD shaper vs. adelay of 0.144 s with ZV-ZV shaper) so that the non-robust ZV-shapers are faster (Fig. 12a) than the robustZVD-shapers.Besides vibration reduction, the input-shaping reduces the control effort. Figure 14 shows cable torque set-pointalong the shaped trajectory with and without input-shaping. The torque set-point of the first (second, third, resp.)cable is equal to 0.138 N.m (0.272 N.m, 0.160 N.m, resp.) at start time when no input-shaping is applied. It isequal to 0.134 N.m (0.263 N.m, 0.155 N.m) when ZV-ZV shaping is applied, which presents a torque reduction of2.89 % (3.3%, 3.12 %, resp.). This value is equal to 0.133 N.m (0.261 N.m, 0.154 N.m, resp.) when ZVD-ZVDinput-shaping is applied, which presents a torque reduction of 3.62 % (4.04%, 3.75 %, resp.) with respect to thenon modified torque. The torque set-point of the first (second, third, resp.) cable is equal to 0.210 N.m (0.385 N.m,0.228 N.m, resp.) at the end of transitional phase when no input-shaping is applied. It is equal to 0.200 N.m(0.367 N.m, 0.226 N.m, resp) when ZV-ZV shaping is applied, which presents a torque reduction of 4.76 % (4.67 %,0.88 %, resp.). This value is equal to 0.156 N.m (0.298 N.m, 0.181 N.m, resp.) when ZVD-ZVD input-shaping isapplied, which presents a torque reduction of 25.71 % (22.59%, 20.61 %, resp.) with respect to the non modifiedtorque.
It is clear that the use of input-shapers leads to better attenuation of residual vibrations, especially while usingrobust input-shapers. However, these vibrations are not totally attenuated.It is noteworthy that an important part of damping comes from the actuators. The non consideration of thesedamping parameters may degrade partially the effectiveness of input-shaping filters.15 a) (b)(c)
Fig. 14: Cable torque set-point without IS, with ZV-ZV shaper and with ZVD-ZVD shaper: (a) cable 1, (b) cable 2,(c) cable 3Note that the choice of constant modal parameters for the input-shapers in this paper was validated by arobustness analysis. This is not true for all CDPR geometric configurations and/or payloads. The degradation ofinput-shaper performance can be caused by mechanical parameters and payload uncertainties and/or variationsdue to evolution over the workspace [68]. In this case, adaptive input-shapers as proposed in [65, 66, 67] should beused in order to reduce uncertainties in modal parameters that lead to residual vibrations.As shown in Fig. 15, the use of input-shapers into the feed-forward control degrades the moving-platformtrajectory tracking due to the delays and impulses applied to the original control signal. To deal with this issue,it would be better to combine the control scheme proposed in this paper with the work presented in [33]. Thislatter focuses on the compensation of the moving-platform deflection coming not only from the elastic behavior ofcables but also from the moving-platform dynamics. The efficiency of such a control scheme in terms of trajectorytracking improvement has been experimentally validated for a non redundant and suspended CDPR [33].
This paper proposed a frequency-dependent method to attenuate the unwanted vibrations of Cable-DrivenParallel Robots (CDPRs). This method deals with the integration of input-shaping filters into the closed-loopfeed-forward control. Two classes of input-shapers were proposed: single-mode and multi-mode input-shapers.These filters re-design the input signal to cancel residual vibrations. A new type of workspace related to theshaper sensitivity to the variations in natural frequencies was defined. This workspace aims to verify the shaperperformance along a prescribed trajectory based on its starting point. The comparison between the velocity errors16
Time (s) -0.06-0.05-0.04-0.03-0.02-0.0100.01 δ z ( m ) Fig. 15: Experimental results: Moving-platform position error along z-axis with and without input-shapingobtained through experimentations when using unshaped input signal or shaped ones as control references showsmeaningful differences. Accuracy improvement with respect to the unshaped control in terms of Peak-to-Peakamplitude of velocity error achieves 53 % while using the ZVD input-shaper and 72 % while using the ZVD-ZVDinput-shaper. This percentage is equal to 36 % with a ZV input-shaper and to 42 % with a ZV-ZV input-shaper.Experimental results confirm that the integration of input-shaping filters into the closed-loop feed-forward controlscheme is relevant to attenuate residual vibrations of a non-redundant CDPR. Further experiments will be performedwhile compensating cable elongations and considering redundantly actuated CDPRs later on. Integrating anadaptive input-shaper into the feed-forward control should further improve the overall CDPR performance. Inorder to improve the performance of multi-mode input-shaping control schemes, an on-line estimation of CDPRdamped natural frequencies, with respect to CDPR geometry, cable tensions and the moving-platform trajectory,will allow the user to tune input-shaping filters as a function of those estimated along a prescribed trajectory of themoving-platform and to reduce the moving-platform pose stabilization time.
DOF : Degree-Of-Freedom.CDPR : Cable-Driven Parallel Robot.ZV : Zero-Vibration.ZVD : Zero-Vibration-Derivative. F b = { O, x b , y b , z b } : Base frame. F p = { P, x p , y p , z p } : Moving-platform frame. n : Number of cables. m : Total number of DOF. K p : Proportional gain of PID controller. K i : Integral gain of PID controller. K d : Derivative gain of PID controller. f : Natural frequency. ζ : Damping ratio. a i : Cartesian coordinate vector of anchor points A i expressed in F p . b i : Cartesian coordinate vector of exit points B i expressed in F b . x : Moving-platform pose vector expressed in F b . p : Moving-platform position vector. o : Moving-platform orientation vector. t : Moving-platform twist. 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