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Trajectory planning of soft link robots with improved intrinsic safety*
Proceedings of the 20th World Congress Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France,Federation July 9-14, 2017 The International of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 6016–6021
Trajectory planning of soft Trajectory planning of soft Trajectory planning of soft Trajectory planning of soft improved intrinsic improved intrinsic improved intrinsic improved intrinsic ∗
link robots link robots link robots link robots safety safety safety safety ∗∗
with with with with
∗ Arthur u Arthur Lismonde Lismonde ∗ Valentin Valentin Sonneville Sonneville ∗∗ Olivier Olivier Br¨ Br¨ uls ls ∗∗ Arthur Lismonde ∗∗ Valentin Sonneville ∗∗ Olivier Br¨ u ls Arthur Lismonde Valentin Sonneville ∗∗ Olivier Br¨ uls ∗ ∗ ∗ Department of Aerospace & Mechanical engineering, University of Department of Aerospace & Mechanical engineering, University of ∗ of Aerospace & Mechanical engineering, University of Li` eege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). ∗ Department Li` ge, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). Department of Aerospace & Mechanical engineering, University of ∗∗ Li` ege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). Engineering, University of ∗∗ Aerospace Engineering, University of Maryland, Maryland, USA USA (e-mail: (e-mail: Li`ege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). ∗∗ Aerospace University of Maryland, USA (e-mail: [email protected]) ∗∗ Aerospace Engineering, [email protected]) Aerospace Engineering, University of Maryland, USA (e-mail: [email protected]) [email protected])
Abstract: Abstract: For For human-robot human-robot cooperation, cooperation, intrinsic intrinsic safety safety approaches approaches aim aim at at reducing reducing the the energy energy Abstract: For human-robot cooperation, intrinsic safety approaches aim at compliance reducing theinenergy involved in the motion of the robotic system and at increasing the system order involved in For the human-robot motion of thecooperation, robotic system and at increasing the system order Abstract: intrinsic safety approaches aim at compliance reducing theinenergy involved inthe therisk motion of the robotic system and at increasing the system compliance in order to reduce of injury in case of an unexpected collision. Robots based on lightweight and to reduceinthe of injury in robotic case of an unexpected collision. Robots basedcompliance on lightweight and involved therisk motion of the system and at increasing the system in order to reduce the risk of injury in case of an unexpected collision. Robots based on lightweight and inherently flexible links exhibit attractive features in this context but the control of their motion inherently flexible links exhibit attractive features in this context but the control of their motion to reduce the risk of injury in case of an unexpected collision. Robots based on lightweight and inherently flexible links exhibit attractive features in this context buttrajectory the control of theirmethod motion then leads to a tremendous challenge. This paper presents a novel planning then leads flexible to a tremendous challenge. This paper in presents a novel planning inherently links exhibit attractive features this context buttrajectory the control of theirmethod motion then3D leads to awhich tremendous challenge. This paper presents a novel trajectory planning method for robots aims improving the accuracy despite the flexibility. for robots aims at atchallenge. improvingThis the tracking tracking accuracy despite the link link planning flexibility.method then3D leads to awhich tremendous paper presents a novel trajectory for 3D robots which aims at improving the tracking accuracy despite the link flexibility. for 3D robots which aims Federation at improving the tracking accuracy the Ltd. link All flexibility. © 2017, IFAC (International of Automatic Control) Hostingdespite by Elsevier rights reserved. Keywords: Keywords: Tracking, Tracking, Control Control of of constrained constrained systems, systems, Nonlinear Nonlinear predictive predictive control, control, Modeling Modeling Keywords: Tracking, Control of constrained systems, Nonlinear predictive control, Modeling for control optimization, Vibration control, Robots for control Tracking, optimization, Vibration control, Modeling, Modeling, Robots manipulators manipulators Keywords: Control of constrained systems, Nonlinear predictive control, Modeling for control optimization, Vibration control, Modeling, Robots manipulators for control optimization, Vibration control, Modeling, Robots manipulators 1. objective 1. INTRODUCTION INTRODUCTION objective is is to to allow allow the the manipulator manipulator to to track track accurately accurately 1. INTRODUCTION objective is to allow theinmanipulator to track accurately a trajectory prescribed time. The feedforward input aobjective trajectory prescribed time. The feedforward input is is 1. INTRODUCTION is to allow theinmanipulator to track accurately acomputed trajectory prescribed ininverse time. The feedforward input is by solving the dynamics of the flexible Human-robot interaction will probably be omnipresent in byprescribed solving theininverse dynamics of the input flexible a trajectory time. The feedforward is Human-robot interaction will probably be omnipresent in computed by on solving the inverse dynamics ofsystems, the flexible MBS based its model. For fully rigid the Human-robot interaction will probably beinterest omnipresent in computed the industry in the near future. Research on safer based its model. For fully rigid ofsystems, the computed by on solving the inverse dynamics the flexible the industry ininteraction the near future. Research on safer Human-robot will probably beinterest omnipresent in MBS MBS based on its model.could For lead fullytorigid systems, computed torque method this In the industry in the near future. Research interest on safer robot is By increasing their compliance and torque method this result. result. In the the MBS based on its model.could For lead fullytorigid systems, robot is spreading. spreading. By future. increasing their interest compliance and computed the industry in the near Research on safer torque method could lead to internal this result. In the case of underactuated systems, some dynamics robot is spreading. By increasing their compliance and computed using lightweight structures, robotic systems can become case of underactuated systems, some internal dynamics computed torque method could lead to this result. In the using structures, robotictheir systems can become robot lightweight is spreading. By increasing compliance and case of underactuated systems, when some the internal dynamics has considered. system is using lightweight structures, robotic systemsincan become intrinsically safe less is their motoofbe beunderactuated considered. Especially Especially system is nonnoncase to systems, when some the internal dynamics intrinsically safe as as less energy energy is involved involved their mo- has using lightweight structures, robotic systemsincan become has to be considered. Especially when the issystem is nonminimum i.e. internal unstable, the intrinsically safelightweight as less energy is involved in their motion. Although robots flexible links can phase, i.e. the the internal dynamics dynamics unstable, the has to be phase, considered. Especially when the issystem is nontion. Although robotsis with with flexible links mocan minimum intrinsically safelightweight as less energy involved in their minimum phase, i.e. the internal dynamics is unstable, the solution of the inverse dynamics can end up unbounded. tion. Although lightweight robots with flexible links can exhibit attractive features in this context, their dynamic solution of the inverse dynamics can end up unbounded. minimum phase, i.e. the internal dynamics is unstable, the exhibit attractive features in this context, their links dynamic tion. Although lightweight robots with flexible can solution of athebounded inverse dynamics can end upBook unbounded. obtain solution, Kwon and (1994) exhibit attractive features in this to context, their dynamic To behavior can be more challenging manage as vibration To obtain a bounded solution, Kwon and Book (1994) solution of the inverse dynamics can end up unbounded. behavior can be more challenging manage as vibration exhibit attractive features in this to context, their dynamic To obtain the aa bounded Kwon and Book dynamics of system causal behavior can be morecan challenging to manage as vibration separates and flexibility issues arise. it dynamicssolution, of aa linear linear system into a (1994) causal To obtain the bounded solution, Kwon and into Booka (1994) and flexibility issues arise. Therefore, Therefore, it is isasimportant important behavior can be morecan challenging to manage vibration separates separates the dynamics ofbefore a linear system into a causal and an anti-causal part solving the inverse dyand flexibility issues can arise. Therefore, itprocess is important to consider such phenomena in the control of the an anti-causal partofbefore solving theinto inverse dyseparates the dynamics a linear system a causal to such phenomena in Therefore, the controlitprocess of the and andconsider flexibility issues can arise. is important an this anti-causal part before solving thetime inverse dynamics, results in aa method called the domain to consider such phenomena in the control process of the and system. namics, this results in method called the time domain and an anti-causal part before solving the inverse dysystem. to consider such phenomena in the control process of the namics, this results in causal a method called the leads time to domain inverse dynamics. The characteristic some system. Flexible robotic arms are multibody systems (MBS) that inverse dynamics. The causal characteristic leads to some namics, this results in a method called the time domain Flexible robotic arms are multibody systems (MBS) that system. dynamics. Theresulting causal characteristic leads to some postactuation in input anti-causal Flexible robotic arms are multibody systems (MBS) that inverse are characterized by a number of in the the input while while the the anti-causal inverse dynamics. Theresulting causal characteristic leads to some are characterized by having having a smaller smallersystems number(MBS) of control control Flexible robotic arms are multibody that postactuation postactuation in thetoresulting input while the system. anti-causal characteristic leads some preactuation in By are characterized by of having a smaller number ofthey control inputs than degrees freedom (dof), meaning are some preactuation in the system. By postactuation leads in thetoresulting input while anti-causal inputs than degrees freedom (dof), number meaningofthey are characteristic are characterized by of having a smaller control leads to contributions, some preactuation in the system. By combining these two the inverse dynamics inputs than degrees of control freedomon(dof), meaning they area characteristic underactuated. To take the unactuated dofs, combining these two contributions, the inverse dynamics characteristic leads to some preactuation in the system. By underactuated. To take the unactuated dofs,area combining these two contributions, the inverse dynamics inputs than degrees of control freedomon(dof), meaning they the flexible system has a non-causal solution. In the case underactuated. To take control onusing the unactuated dofs, a of monitoring system of the system sensors can be set of the flexible system has a non-causal solution. In the case combining these two contributions, the inverse dynamics monitoring system of the system using sensors can be set underactuated. To take control on the unactuated dofs, a of the flexiblesystems, system has a non-causal solution. introduced In the case nonlinear stable inversion monitoring system of the system using sensors cantobe set of the and suitable action can be fed to system comnonlinear stable inversion methods methods flexiblesystems, system has a non-causal solution. introduced In the case and suitablesystem action of canthe besystem fed back back to the the system commonitoring using sensors cantobe set by of nonlinear systems, stable inversion methods introduced Devasia et al. (1996) are necessary. Their application and suitable action canmotion be fed back to the system to com- by pensate for (see Cannon and Devasia et al. (1996) areinversion necessary. Their application nonlinear systems, stable methods introduced pensate for undesired undesired (see to Cannon and Schmitz Schmitz and suitable action canmotion be fed back the system to com- of by Devasia et al. and (1996) are necessary. Their application to flexible MBS by pensate for undesired motion (see Cannon and Schmitz (1984)). Vibrations and elastic deformations can also underactuated flexible MBS is is described described by Seifried Seifried by underactuated Devasia et al. and (1996) are necessary. Their application (1984)). Vibrations andmotion elastic(see deformations can Schmitz also be be to pensate for undesired Cannon and underactuated andEberhard flexible MBS is described by Seifried (2014); Seifried and (2009). In this case, the (1984)). Vibrations and elasticof deformations can aalso be to predicted based on an analysis the system. With modal Seifried and (2009). In this case, the ininto underactuated andEberhard flexible MBS is described by Seifried predicted based on anand analysis the system. With modal (1984)). Vibrations elasticof deformations can aalso be (2014); Seifried problem and Eberhard (2009). In this case, the inverse dynamics is stated as a two-point boundary predicted based on an analysis of the system. With ashaping modal (2014); analysis, Singer and Seering (1990) used input dynamics is stated as a two-point boundary (2014); Seifried problem and Eberhard (2009). In this case, the inanalysis, and Seering of (1990) used input predicted Singer based on an analysis the system. With ashaping modal verse verse dynamics problem is stated as a two-point boundary value problem initial and final conditions have analysis, Singer and Seering (1990)ofused input shaping methods cancel out the the at problem where where initial and as final conditionsboundary have to to verse dynamics problem is stated a two-point methods to canceland out Seering the vibrations vibrations the system system at some some value analysis, to Singer (1990)ofused input shaping value problem where initial and final conditions have to be met. The final condition, assures that the system ends methods to cancel out the vibrations of the system at some preselected frequencies. Staufer and Gattringer (2013) on be met. The final condition, assures that the system ends value problem where initial and final conditions have to preselected frequencies. Staufer and of Gattringer (2013) on be met. The final condition, assures that the system ends methods to cancel out the vibrations the system at some a stable manifold, i.e. a configuration space where all preselected frequencies. Stauferelement and Gattringer (2013) on on the other hand, used a lumped model model of the a stable i.e. a configuration space where all be met. Themanifold, final condition, assures that the system ends the other hand, used a lumped model model of the preselected frequencies. Stauferelement and Gattringer (2013) on on a stable manifold,toi.e. a configuration space wheregoes all trajectories aa given configuration as the other hand, usedsuitable a lumped element model model of the on system to compute control inputs. These feedback converge given configuration as time time on a stable converge manifold,toi.e. a configuration space wheregoes all system to hand, compute control inputs. These feedback the other usedsuitable a lumped element model model of the trajectories trajectories converge to a given configuration as time goes forward. On the other side, the initial condition forces system to compute suitable control inputs. These feedback and feedforward are complementary and can Onconverge the other initial condition trajectories to aside, giventhe configuration as timeforces goes and feedforward actions arecontrol complementary andfeedback can be be forward. system to computeactions suitable inputs. These Ontothe other side, the initial condition forces the system start on unstable manifold, i.e. conand feedforward actions are complementary anddone can by be forward. combined to improve robustness system start on an an unstable manifold, i.e. a aforces conforward. Ontothe other side, the initial condition combined to further further improve robustness as as was was and feedforward actions are complementary anddone can by be the systemspace to start onall antrajectories unstable manifold, i.e. agiven configuration where converge to a combined toLuca further improve robustness as was (2013). done by the for e.g., De (2000); Staufer and Gattringer where converge i.e. to aagiven the systemspace to start onall antrajectories unstable manifold, confor e.g., DetoLuca (2000); Staufer and Gattringer combined further improve robustness as was (2013). done by figuration figuration space where all trajectories converge to aisgiven configuration as time goes backward. This method able for e.g., De Lucaway (2000); Staufer and Gattringer (2013). An innovative to compute the feedforward action as where time goes backward. This method able figuration space all trajectories converge to aisgiven An innovative to compute the Gattringer feedforward action configuration for e.g., De Lucaway (2000); Staufer and (2013). configuration as time goes backward. This method isODE able to solve the inverse dynamics either in the form of an An innovative way to compute the feedforward action of flexible manipulators is presented in this work. The to solve the inverse dynamics either in the form of an ODE configuration as time goes backward. This method is able of manipulators is presented in this work.action The to solve the inverse dynamics either in the form of an ODE Anflexible innovative way to compute the feedforward in the form of a differential-algebraic equation (DAE), of flexible manipulators is presented in this work. The or or in the form of a differential-algebraic equation (DAE), to solve the inverse dynamics either in the form of an ODE of flexible manipulators is presented in this work. The or in the form ofuals differential-algebraic equation (DAE), publication is funded by the European Regional Development as shown by et to the The The publication is funded by the European Regional Development as shown by Br¨ Br¨ et al. al. (2014). (2014). In In order order to avoid avoid the or in the form ofuals differential-algebraic equation (DAE), Fund (ERDF) within European Union’s INTERREG A-program The publication is funded by the European Regional V Development as shown of by boundary Br¨ uls et al. (2014). In order to avoid the definition values, an optimization approach The(ERDF) Fund within European Union’s INTERREG V A-program publication is funded by the European Regional Development definition of boundary values, an optimization approach as shown by Br¨ u ls et al. (2014). In order to avoid the Greater Region,within project Robotix Union’s Academy. Fund (ERDF) European INTERREG V A-program definition of boundary values, an optimization approach was proposed by Bastos et al. (2013). This optimization Greater Region,within project Robotix Union’s Academy. Fund (ERDF) European INTERREG V A-program was proposed by Bastosvalues, et al. an (2013). This optimization definition of boundary optimization approach In addition, the project first author would like to acknowledge the Belgian Greater Region, Robotix Academy. was proposed by Bastos et al. (2013). This optimization method also to solution which converges In addition, the project first author would like to acknowledge the Belgian Greater Region, Robotix Academy. method also leads leads to aa bounded bounded solutionThis which converges was proposed by Bastos et al. (2013). optimization Fund for Research in Industry and Agriculture for its In addition, the first training author would like to acknowledge the Belgian method also leads to atwo-point bounded solution which converges Fund for Research in Industry and Agriculture for its to the solution of the boundary value problem In addition, the first training author would like to acknowledge the Belgian to the solution of the two-point boundary value problem method also leads to a bounded solution which converges financial (FRIA grant). Fund forsupport Research training in Industry and Agriculture for its financial (FRIA grant). to the solution of the two-point boundary value problem Fund forsupport Research training in Industry and Agriculture for its financial support (FRIA grant). to the solution of the two-point boundary value problem financial support (FRIA grant).
Copyright © 2017, 2017 IFAC 6205Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 6205 Copyright 2017 responsibility IFAC 6205Control. Peer review©under of International Federation of Automatic Copyright © 2017 IFAC 6205 10.1016/j.ifacol.2017.08.1440
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Arthur Lismonde et al. / IFAC PapersOnLine 50-1 (2017) 6016–6021
when the length of the pre- and post-actuation phases increase (see Bastos et al. (2017)). Other numerical methods dealing with the inverse dynamics of flexible MBS in DAE form are compared by Moberg and Hanssen (2009). In the end, all methods lead to bounded and non-causal solutions. Here, the focus is given to the inverse dynamics of flexible 3D MBS for trajectory tracking tasks. The optimization formulation is chosen in combination with a finite element model of the system. Although the use of finite elements such as flexible beams, rigid bodies and kinematic joints elements can lead to a high dimensional model, it is a straight forward way to model general MBS, would it be a parallel or a serial system. A finite element method formulated on the special Euclidean group SE(3) is chosen. As shown by Br¨ uls et al. (2011), this formulation can benefit from a local frame formulation that reduces the nonlinearities of the equations of motion. Rotations in space can also be represented without singularity issues. In this paper, the use of finite elements in the SE(3) formalism is shortly described to understand how the flexible arm is modeled. Based on this model, the optimization problem, its objective function and its constraints are formulated. This constrained optimization problem is then solved using a direct transcription method (see Betts (2010)). Eventually, the case of a flexible serial manipulator is analyzed to illustrate this novel trajectory planning method. 2. MODEL OF A FLEXIBLE MULTIBODY SYSTEM 2.1 Finite element model The finite element method is a widespread tool when dealing with MBS. It is indeed a straight forward way to represent rigid bodies, flexible bodies and kinematic joints that compose a robotic arm. Here, the MBS is discretized in space into N nodes and a special Euclidean group SE(3) formalism is used to represent position and rotation. With such formalism, the position and orientation of each node I, with I = 1, ..., N , are represented as a homogeneous 4×4 matrix HI with a rotation RI ∈ SO(3) and a position pI ∈ R3 component. RI p I ∈ SE(3) HI = 0 1 The model is then constructed by connecting each node with rigid bodies, flexible bodies and kinematic joints (see Sonneville and Br¨ uls (2014); Sonneville et al. (2014)). In this work, flexible link are modeled using beam elements. Figure 1 gives an example of the discretization of a serial arm into N = 10 nodes connected with a rigid body, 4 beam elements and 3 kinematic joints. This example is further described and analyzed in the next section. Soft joints could also be considered by adding spring-damper parameters in the kinematic joints elements, but this is not considered here. 2.2 Equations of motion For trajectory planning and tracking problems, the endeffector position yef f of the flexible arm has to follow a prescribed path ypresc (t), evolving in time and defined by r scalar components. This can be achieved thanks to r
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Fig. 1. Discretization into N = 10 nodes of a flexible serial arm control inputs u = (u1 , ..., ur ). The dynamics of such MBS is governed by ˙ I = HI v ˜I (1) H Mv˙ + g(H, v) + BT λ = Au (2) Φ (H) = 0 (3) yef f (H) − ypresc (t) = 0 (4) In this set of equations, H = diag(H1 , ..., HN ) is the nodal configuration that gathers each nodal variable HI T T with I = 1, ..., N and v = (vIT , ..., vN ) is the vector of nodal velocities. The matrix M is the symmetric mass matrix of the system, g is the vector of internal and complementary inertia forces, B is the gradient of the m kinematic constraints Φ , which represent the connections imposed by the kinematic joints. Related to these kinematic constraints is the m dimensional vector λ of Lagrange multipliers. The matrix A is a boolean matrix applying the controls u to the system. The last equation is called the servo constraint (see Blajer and Kolodziejczyk (2004)) and compels the end-effector yef f to follow the prescribed trajectory ypresc (t). Flexible systems are said to be underactuated, in that case, the system has some internal dynamics of dimension 6N − m − r. With initial conditions on this internal dynamics, the behavior of the system would be completely defined. Unfortunately, the forward propagation of such initial conditions would lead to unbounded control inputs u requirements which are unachievable in reality. In order to find a bounded solution, no initial conditions of the system are defined and the inverse dynamics problem is formulated as an optimization problem. 3. OPTIMIZATION PROBLEM 3.1 Formulation As already mentioned previously, stable inversion techniques have to be used in order to have a bounded solution when solving the inverse dynamics of non-minimum phase systems. One possibility is to formulate the inverse problem as a constrained optimization problem. In our case, the objective is to minimize the internal dynamics of our flexible system. Considering that the internal dynamics can be represented using a function φ(H) depending on the nodal configurations, the optimization problem minimizes the objective function J on the given time lap T = tf − ti and can be mathematically written as tf 1 min J = min φ(H)dt (5) H H 2T t i This minimization has to be carried out by considering the constraints defined by the equations of motion of
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the flexible MBS given by (1)-(4) for t ∈ [ti , tf ]. In this formulation, no initial and final conditions on H and v are set and they are to be determined by the optimization process.
linked to each nodal configuration by (10) and involves the exponential mapping expSE(3) (˜•) that maps a 6×1 vector to an element of SE(3). k ) Hk+1 = Hk exp (∆Q (10) I
I
SE(3)
I
3.2 Direct transcription 3.4 Optimization variables To initialize the optimization, the initial guess of the variables, denoted with a hat ˆ •, are computed by solving the inverse dynamics of the equivalent rigid system. This rigid system can easily be solved using a computed torque method as, in this case, the system would have no internal dynamics. To solve the optimization problem, the direct transcription method is used and the evolution of the system is discretized into s time steps tk , with k = 1, ..., s, resulting in a discrete Nonlinear Program (NLP). Fig. 2 illustrates the direct transcription method for the configuration H.
After discretization, the unknown variables of the optimization problem are (H1 , v1 , v˙ 1 , a1 , λ 1 , u1 , ..., Hs , vs , v˙ s , as , λ s , us ) with Hk = diag(Hk1 , ..., HkN ) for k = 1, ..., s. The nodal variables HkI ∈ SE(3) belong to the SE(3) group and can therefore not be treated directly by classical NLP algorithms. To overcome this issue, an incremental configuration variable ∆q = (∆qT1 , ..., ∆qTN )T defined on a tangential space of SE(3) is introduced (see also Absil et al. (2008)). This new incremental variable determines ˆ and the current the variation between the initial guess H value H. At time step k, the relation between the configuration Hk = (Hk1 , ..., HkN ) and the incremental variable ∆qk = (∆qk1 , ..., ∆qkN ) is k ) ˆ k exp (11) (∆q Hk = H I
Fig. 2. Direct transcription method: optimization starting ˆ s ) that leads to the ˆ 1 , ..., H from the initial guess (H 1 s optimal trajectory (H , ..., H )
I
SE(3)
I
It is important to notice the difference between (10) and (11). The former relates the nodal configuration of the system at the same iteration but at two consecutive time steps, i.e. Hk and Hk+1 . The latter relates the current nodal configuration of the system to its initial guess at a ˆ k and Hk . Figure 3 is a zoom of Fig. 2 given time step, i.e. H and illustrates the relation between these variables where each arrow is an exponential mapping expSE(3) with either time incremental or iteration incremental arguments, i.e. ∆Qk and ∆qk respectively.
As the system is now discretized in time, the objective function written in its discrete form is now given by s 1 φ(Hk )h (6) J= 2T k=1
with h being the time step size. 3.3 Time integration scheme
In addition to the motion constraints given by (1)-(4), some time constraint must be considered in the discrete problem. The Lie group generalized-α integration scheme (see Br¨ uls et al. (2012)) is used here as it is able to deal with the equations of motion formulated in the Lie group SE(3). At each time step tk , these relations must be satisfied 1 ∆Qk − hvk − ( − β)h2 ak − βh2 ak+1 = 0 (7) 2 vk+1 − vk − (1 − γ)hak − γhak+1 = 0 (8) (1 − αm )ak+1 + αm ak − (1 − αf )v˙ k+1 − αf v˙ k = 0 (9)
where ak can be seen as an estimation of the acceleration, β, γ, αm and αf are parameters of the method, defining its behavior. The time incremental variable ∆Qk = k,T (∆Qk,T 1 , ..., ∆QN ) represents the incremental motion made by each node between two consecutive time steps tk and tk+1 . For each node I, this incremental variable is
Fig. 3. Relation between variables ∆qk , ∆Qk and Hk . In the end, the design variables x are x = (∆q1 , v1 , v˙ 1 , a1 , λ 1 , u1 , ..., ∆qs , vs , v˙ s , as , λ s , us ) and the optimization problem has only vectorial variables that can be treated using classical NLP solvers. The feedforward action u(t) can then be reconstructed at any time interpolation between the discrete values u1 , ..., us . 3.5 Optimization solver The combination of the finite element formalism and the direct transcription method leads to a large optimization problem with a high number of design variables and constraints. However, the problem is sparse as coupling
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terms are limited to consecutive time steps (see (7)-(10)) and nodes directly connected in the finite element mesh. Solvers such as KNITRO, IPOPT and FMINCON are able to deal with large scale and sparse problems. With appropriate options, the FMINCON solver in Matlab performed better for the tested cases and was chosen in the present work. To improve computation time, the gradients of the objective function J and the constraint equations are computed analytically and provided to the solver. 4. SERIAL MANIPULATOR EXAMPLE To illustrate the impact that can be done on the tracking precision by considering the flexible behavior of the MBS, let us consider the case of a flexible 3D serial manipulator. This manipulator has 3 actuated joints and two links as shown in Fig. 4. The first link has length l and a tubular square cross section of outer side a1 and edge thickness e1 . The second link also has length l but has a smaller tubular square cross section of side a2 and edge thickness e2 . The ratio between side a2 and thickness e2 of the second link is equal to the ratio a1 /e1 of the first link. In the initial configuration, each link is making a 45◦ angle with the horizontal plane and the input torques u1 ,u2 and u3 have their axis along z, y and y respectively. The point mass end-effector, with mass mend , is controlled using these three torque inputs. Flexibility is only considered in the second link and is modeled using 4 beam elements. The description of the beam formulation on SE(3) can be found in Sonneville et al. (2014).
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properties. A high numerical damping is considered for the optimization process; a spectral radius of ρ∞ = 0.01 is chosen (β = 0.98, γ = 1.48, αm = −0.97 and αf = 0.01). It is important to note that the convergence of the optimization process is quite sensitive to the choice of the initial guess. When gravity is acting on the system, the convergence is better when the initial guess starts with the static deflection of the arm. In this example, the integrand φ(Hk ) is defined as the strain energy of the flexible beams of the system. The number of time steps s is chosen in order to capture best the dynamics of the system: the inverse of the time step size 1/h is chosen to be five times higher than the first unstable frequency of the flexible system. For this serial arm, the first unstable frequency is 13 Hz and s = 150 steps is chosen, leading to 1/h = 100. This results in a optimization problem with around 39k variables that takes 5 iterations and around 2 minutes to solve (using a x64 bits i7-4600u CPU with 16 Gb RAM memory). For this case, Fig. 5 shows the input commands u and urigid , inputs with and without flexibility considerations respectively. Although some differences can be seen between these two control inputs, the pre-or post-actuation phases are not quite visible. By looking at the velocities of the joints, one can see in Fig. 6 that the velocity of joint 3 is still varying after the 1.3 s. Although the post-actuation torque is quite small, it still result in a motion of the joints of the system, i.e. internal motion of the system, during this post-actuation phase. Similar observations can be done for the pre-actuation phase. It is important to note that while this internal motion is present, the end-effector itself does not actually move during these pre- and post-actuation phases.
Commands [Nm]
10
Fig. 4. Serial 3D arm system with one rigid body and 4 beam elements.
Link 2 End-eff.
l=1m
a1 = 0.05 m
e1 = 0.01 m ρ = 2700 kg/m3
l=1m E = 70 GPa
a2 = 0.0075 m ν = 0.3
e2 = 0.0015 m ρ = 2700 kg/m3
u1 u2
−20
u3 u1,rigid u2,rigid u3,rigid 0
0.2
0.5
1
1.3
1.5
Time [s]
Fig. 5. Input commands of the flexible 3D arm. 2
Table 1. Parameters of the serial arm system. Link 1
−10
−30
Joints velocity (rad/s)
The trajectory that has to be tracked is a planar circular arc of diameter d in the yz plane. The end-effector starts from position (2l cos(45◦ ), 0, 0) and goes to position (2l cos(45◦ ), d, 0). The motion profile is built using a seventh order polynomial to insure continuity of the positions, velocities, accelerations and jerks over time. The circular arc trajectory is covered in 1.1 s an the pre- and postactuation phases both last 0.2 s: the total simulation time is 1.5 s. Table 1 shows the material and geometrical parameters of the serial manipulator.
0
Joint 1 Joint 2 Joint 3 1
0
−1
mend = 0.1 kg −2
The initial guess is computed using a complete rigid model of the system. The second link is temporarily considered as a rigid body with the same geometrical and material
0
0.2
0.5
1
1.3
Time (s)
Fig. 6. Joint velocities of the flexible 3D arm.
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Once the inverse dynamics is solved, the resulting optimized input torques u are applied to the system to perform a first direct dynamic analysis. In parallel, a second direct dynamic analysis is performed using the input torques urigid . These input commands result in an end-effector trajectory shown in Fig. 7. The relative tracking error resulting from both direct simulation can be calculated. For each time step tk , this relative error can be calculated using k k − yef ypresc f 2 (12) ek = k ypresc 2
where • 2 is the classical Euclidean norm or L2 norm and k = 1, ..., s. The relative rms error is calculated as s 1 (ek )2 (13) erms = s
Fig. 8. Evolution of the tracking error with respect to a2 . 5. CONCLUSION
k=1
Using the optimized inputs u, the relative rms error erms done on the trajectory drops down from 1.1% to 0.25%.
Y [m]
1
With u With urigid Prescribed
0
1.39
1.395
1.4
1.405
1.41
1.415
1.42
X [m]
Fig. 7. End-effector trajectory of the flexible 3D arm using u and urigid . For a given design of the serial manipulator, considering flexibility in the manipulator arm obviously result in a better tracking precision. This improvement can lead to a more lightweight design of the manipulator. We now consider a similar serial manipulator for which the side length a2 of the flexible second link is a parameter that will vary from amax = 0.1 m to amin = 0.0075 m. For each value of a2 , the control inputs that take into account flexibility and those where flexibility is not taken into account are both applied to the system. In order to be able to compare the tracking precision resulting from each inputs, an external perturbation force is added to the system’s end-effector during the direct dynamic analysis. For the results shown in Fig. 8, the perturbation is represented as an acceleration vector gp = (−0.3, −0.3, 0.3) m/s2 that has an equal magnitude along the x, y and z axes and that acts on the end-effector mass mend . It can be observed that for a relative tracking tolerance of 1.5%, the flexible control allows the design of a second link that has a side length of around 0.025 m instead of 0.05 m. For this particular example, the size of the second arm can be reduced by about half its size and still achieve similar performance as a rigidly controlled arm. The proposed feedforward control technique can thus be exploited to reduce the weight of the moving links and to increase their compliance while preserving the accuracy requirement.
Controlling the flexible behavior of lightweight arms can be challenging. Here, the improvement on the tracking precision coming from considering the flexible behavior of a manipulator are shown for a serial manipulator. A finite element model of the considered flexible MBS is first built using flexible beam elements, rigid bodies and kinematic joints. The inverse dynamics problem is then formulated as a constrained optimization problem where the objective is to minimize the internal dynamics of the flexible system. The use of a SE(3) formalism in the finite element model benefits from a local frame representation that reduces the nonlinearities of the constraint equations. The system is discretized in time and the direct transcription method is used to solve the optimization problem. The example of a serial link manipulator is discussed to show the improvement that can be done when flexibility is considered in the control scheme. By allowing more compliance in the manipulator, these adaptations would contribute to an increased safety of the machine as they induce a significant decrease in the severity of collisions in a human-robot cooperation application. In further works, a feedback loop could be added in the simulation to further analyze the impact of this feedforward command on the control of flexible manipulators. To reduce computation time, reduced order models based on a superelement formulation described by Sonneville (2015) can be considered to reduce the dimension of the optimization problem. REFERENCES Absil, P.A., Mahony, R., and Sepulchre, R. (2008). Optimization algorithms on Matrix Manifolds. Princeton University Press. Bastos, G.J., Seifried, R., and Br¨ uls, O. (2013). Inverse dynamics of serial and parallel underactuated multibody systems using a DAE optimal control approach. Multibody System Dynamics, 30, 359–376. doi: 10.1007/s11044-013-9361-z. Bastos, G.J., Seifried, R., and Br¨ uls, O. (2017). Analysis of stable model inversion methods for constrained underactuated mechanical systems. Mechanism and Machine Theory, 111, 99–117. Betts, J.T. (2010). Practical Method for Optimal Control and Estimation Using Nonlinear Programming. Advances in Desgin and Control. SIAM.
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Blajer, W. and Kolodziejczyk, K. (2004). A geometric approach to solving problems of control constraints: Theory and a dae framework. Multibody System Dynamics, 11, 343 – 364. Br¨ uls, O., Arnold, M., and Cardona, A. (2011). Two Lie group formulations for dynamic multibody systems with large rotations. In Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Washington D.C., U.S. Br¨ uls, O., Bastos, G.J., and Seifried, R. (2014). A stable inversion method for feedforward control of constrained flexible multibody systems. Journal of Computational and Nonlinear Dynamics, 9. doi:10.1115/1..4025476. Br¨ uls, O., Cardona, A., and Arnold, M. (2012). Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory, 48, 121–137. Cannon, R. and Schmitz, E. (1984). Initial experiments on the end-point control of a flexible one-link robot. The International Journal of Robotics Research, 3(3), 62–75. De Luca, A. (2000). Feedforward/feedback laws for the control of flexible robots. In Proceedings of the IEEE International Conference on Robotics & Automation. Devasia, S., Chen, D., and Paden, B. (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control, 41(7). Kwon, D.S. and Book, W.J. (1994). A time-domain inverse dynamic tracking control of a single link flexible manipulator. Journal of Dynamic Systems, Measurement and Control, 116, 193–200. Moberg, S. and Hanssen, S. (2009). Inverse dynamics of flexible manipulators. In Proceedings of the Multibody dynamics, ECCOMAS Thematic Conference. Seifried, R. (2014). Dynamics of Underactuated Multibody Systems: Modeling, Control and Optimal Design. Solid Mechanics and its Applications. Springer. Seifried, R. and Eberhard, P. (2009). Design of feedforward control for underactuated multibody systems with kinematic redundancy. Motion and Vibration Control: Selected papers from MOVIC 2008. Singer, N. and Seering, W.P. (1990). Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement and Control, 112, 76–82. doi: 10.1115/1.2894142. Sonneville, V. (2015). A geometric local frame approach for flexible multibody systems. Ph.D. thesis, University of Li`ege. Sonneville, V. and Br¨ uls, O. (2014). A formulation on the special Euclidean group for dynamic analysis of multibody systems. Journal of Computational and Nonlinear Dynamics, 9. doi:10.1115/1.4026569. Sonneville, V., Cardona, A., and Br¨ uls, O. (2014). Geometrically exact beam finite element formulated on the special euclidean group SE(3). Computer Methods in Applied Mechanics & Engineering, 268, 451–474. doi: 10.1016/j.cma.2013.10.008. Staufer, P. and Gattringer, H. (2013). Multibody System Dynamics, Robotic and Control, chapter PassivityBased Tracking Control of a Flexible Link Robot, 95– 112. Springer Vienna.
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Trajectory planning of soft Trajectory planning of soft Trajectory planning of soft Trajectory planning of soft improved intrinsic improved intrinsic improved intrinsic improved intrinsic ∗
link robots link robots link robots link robots safety safety safety safety ∗∗
with with with with
∗ Arthur u Arthur Lismonde Lismonde ∗ Valentin Valentin Sonneville Sonneville ∗∗ Olivier Olivier Br¨ Br¨ uls ls ∗∗ Arthur Lismonde ∗∗ Valentin Sonneville ∗∗ Olivier Br¨ u ls Arthur Lismonde Valentin Sonneville ∗∗ Olivier Br¨ uls ∗ ∗ ∗ Department of Aerospace & Mechanical engineering, University of Department of Aerospace & Mechanical engineering, University of ∗ of Aerospace & Mechanical engineering, University of Li` eege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). ∗ Department Li` ge, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). Department of Aerospace & Mechanical engineering, University of ∗∗ Li` ege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). Engineering, University of ∗∗ Aerospace Engineering, University of Maryland, Maryland, USA USA (e-mail: (e-mail: Li`ege, Belgium (e-mail: {alismonde,o.bruls}@ulg.ac.be). ∗∗ Aerospace University of Maryland, USA (e-mail: [email protected]) ∗∗ Aerospace Engineering, [email protected]) Aerospace Engineering, University of Maryland, USA (e-mail: [email protected]) [email protected])
Abstract: Abstract: For For human-robot human-robot cooperation, cooperation, intrinsic intrinsic safety safety approaches approaches aim aim at at reducing reducing the the energy energy Abstract: For human-robot cooperation, intrinsic safety approaches aim at compliance reducing theinenergy involved in the motion of the robotic system and at increasing the system order involved in For the human-robot motion of thecooperation, robotic system and at increasing the system order Abstract: intrinsic safety approaches aim at compliance reducing theinenergy involved inthe therisk motion of the robotic system and at increasing the system compliance in order to reduce of injury in case of an unexpected collision. Robots based on lightweight and to reduceinthe of injury in robotic case of an unexpected collision. Robots basedcompliance on lightweight and involved therisk motion of the system and at increasing the system in order to reduce the risk of injury in case of an unexpected collision. Robots based on lightweight and inherently flexible links exhibit attractive features in this context but the control of their motion inherently flexible links exhibit attractive features in this context but the control of their motion to reduce the risk of injury in case of an unexpected collision. Robots based on lightweight and inherently flexible links exhibit attractive features in this context buttrajectory the control of theirmethod motion then leads to a tremendous challenge. This paper presents a novel planning then leads flexible to a tremendous challenge. This paper in presents a novel planning inherently links exhibit attractive features this context buttrajectory the control of theirmethod motion then3D leads to awhich tremendous challenge. This paper presents a novel trajectory planning method for robots aims improving the accuracy despite the flexibility. for robots aims at atchallenge. improvingThis the tracking tracking accuracy despite the link link planning flexibility.method then3D leads to awhich tremendous paper presents a novel trajectory for 3D robots which aims at improving the tracking accuracy despite the link flexibility. for 3D robots which aims Federation at improving the tracking accuracy the Ltd. link All flexibility. © 2017, IFAC (International of Automatic Control) Hostingdespite by Elsevier rights reserved. Keywords: Keywords: Tracking, Tracking, Control Control of of constrained constrained systems, systems, Nonlinear Nonlinear predictive predictive control, control, Modeling Modeling Keywords: Tracking, Control of constrained systems, Nonlinear predictive control, Modeling for control optimization, Vibration control, Robots for control Tracking, optimization, Vibration control, Modeling, Modeling, Robots manipulators manipulators Keywords: Control of constrained systems, Nonlinear predictive control, Modeling for control optimization, Vibration control, Modeling, Robots manipulators for control optimization, Vibration control, Modeling, Robots manipulators 1. objective 1. INTRODUCTION INTRODUCTION objective is is to to allow allow the the manipulator manipulator to to track track accurately accurately 1. INTRODUCTION objective is to allow theinmanipulator to track accurately a trajectory prescribed time. The feedforward input aobjective trajectory prescribed time. The feedforward input is is 1. INTRODUCTION is to allow theinmanipulator to track accurately acomputed trajectory prescribed ininverse time. The feedforward input is by solving the dynamics of the flexible Human-robot interaction will probably be omnipresent in byprescribed solving theininverse dynamics of the input flexible a trajectory time. The feedforward is Human-robot interaction will probably be omnipresent in computed by on solving the inverse dynamics ofsystems, the flexible MBS based its model. For fully rigid the Human-robot interaction will probably beinterest omnipresent in computed the industry in the near future. Research on safer based its model. For fully rigid ofsystems, the computed by on solving the inverse dynamics the flexible the industry ininteraction the near future. Research on safer Human-robot will probably beinterest omnipresent in MBS MBS based on its model.could For lead fullytorigid systems, computed torque method this In the industry in the near future. Research interest on safer robot is By increasing their compliance and torque method this result. result. In the the MBS based on its model.could For lead fullytorigid systems, robot is spreading. spreading. By future. increasing their interest compliance and computed the industry in the near Research on safer torque method could lead to internal this result. In the case of underactuated systems, some dynamics robot is spreading. By increasing their compliance and computed using lightweight structures, robotic systems can become case of underactuated systems, some internal dynamics computed torque method could lead to this result. In the using structures, robotictheir systems can become robot lightweight is spreading. By increasing compliance and case of underactuated systems, when some the internal dynamics has considered. system is using lightweight structures, robotic systemsincan become intrinsically safe less is their motoofbe beunderactuated considered. Especially Especially system is nonnoncase to systems, when some the internal dynamics intrinsically safe as as less energy energy is involved involved their mo- has using lightweight structures, robotic systemsincan become has to be considered. Especially when the issystem is nonminimum i.e. internal unstable, the intrinsically safelightweight as less energy is involved in their motion. Although robots flexible links can phase, i.e. the the internal dynamics dynamics unstable, the has to be phase, considered. Especially when the issystem is nontion. Although robotsis with with flexible links mocan minimum intrinsically safelightweight as less energy involved in their minimum phase, i.e. the internal dynamics is unstable, the solution of the inverse dynamics can end up unbounded. tion. Although lightweight robots with flexible links can exhibit attractive features in this context, their dynamic solution of the inverse dynamics can end up unbounded. minimum phase, i.e. the internal dynamics is unstable, the exhibit attractive features in this context, their links dynamic tion. Although lightweight robots with flexible can solution of athebounded inverse dynamics can end upBook unbounded. obtain solution, Kwon and (1994) exhibit attractive features in this to context, their dynamic To behavior can be more challenging manage as vibration To obtain a bounded solution, Kwon and Book (1994) solution of the inverse dynamics can end up unbounded. behavior can be more challenging manage as vibration exhibit attractive features in this to context, their dynamic To obtain the aa bounded Kwon and Book dynamics of system causal behavior can be morecan challenging to manage as vibration separates and flexibility issues arise. it dynamicssolution, of aa linear linear system into a (1994) causal To obtain the bounded solution, Kwon and into Booka (1994) and flexibility issues arise. Therefore, Therefore, it is isasimportant important behavior can be morecan challenging to manage vibration separates separates the dynamics ofbefore a linear system into a causal and an anti-causal part solving the inverse dyand flexibility issues can arise. Therefore, itprocess is important to consider such phenomena in the control of the an anti-causal partofbefore solving theinto inverse dyseparates the dynamics a linear system a causal to such phenomena in Therefore, the controlitprocess of the and andconsider flexibility issues can arise. is important an this anti-causal part before solving thetime inverse dynamics, results in aa method called the domain to consider such phenomena in the control process of the and system. namics, this results in method called the time domain and an anti-causal part before solving the inverse dysystem. to consider such phenomena in the control process of the namics, this results in causal a method called the leads time to domain inverse dynamics. The characteristic some system. Flexible robotic arms are multibody systems (MBS) that inverse dynamics. The causal characteristic leads to some namics, this results in a method called the time domain Flexible robotic arms are multibody systems (MBS) that system. dynamics. Theresulting causal characteristic leads to some postactuation in input anti-causal Flexible robotic arms are multibody systems (MBS) that inverse are characterized by a number of in the the input while while the the anti-causal inverse dynamics. Theresulting causal characteristic leads to some are characterized by having having a smaller smallersystems number(MBS) of control control Flexible robotic arms are multibody that postactuation postactuation in thetoresulting input while the system. anti-causal characteristic leads some preactuation in By are characterized by of having a smaller number ofthey control inputs than degrees freedom (dof), meaning are some preactuation in the system. By postactuation leads in thetoresulting input while anti-causal inputs than degrees freedom (dof), number meaningofthey are characteristic are characterized by of having a smaller control leads to contributions, some preactuation in the system. By combining these two the inverse dynamics inputs than degrees of control freedomon(dof), meaning they area characteristic underactuated. To take the unactuated dofs, combining these two contributions, the inverse dynamics characteristic leads to some preactuation in the system. By underactuated. To take the unactuated dofs,area combining these two contributions, the inverse dynamics inputs than degrees of control freedomon(dof), meaning they the flexible system has a non-causal solution. In the case underactuated. To take control onusing the unactuated dofs, a of monitoring system of the system sensors can be set of the flexible system has a non-causal solution. In the case combining these two contributions, the inverse dynamics monitoring system of the system using sensors can be set underactuated. To take control on the unactuated dofs, a of the flexiblesystems, system has a non-causal solution. introduced In the case nonlinear stable inversion monitoring system of the system using sensors cantobe set of the and suitable action can be fed to system comnonlinear stable inversion methods methods flexiblesystems, system has a non-causal solution. introduced In the case and suitablesystem action of canthe besystem fed back back to the the system commonitoring using sensors cantobe set by of nonlinear systems, stable inversion methods introduced Devasia et al. (1996) are necessary. Their application and suitable action canmotion be fed back to the system to com- by pensate for (see Cannon and Devasia et al. (1996) areinversion necessary. Their application nonlinear systems, stable methods introduced pensate for undesired undesired (see to Cannon and Schmitz Schmitz and suitable action canmotion be fed back the system to com- of by Devasia et al. and (1996) are necessary. Their application to flexible MBS by pensate for undesired motion (see Cannon and Schmitz (1984)). Vibrations and elastic deformations can also underactuated flexible MBS is is described described by Seifried Seifried by underactuated Devasia et al. and (1996) are necessary. Their application (1984)). Vibrations andmotion elastic(see deformations can Schmitz also be be to pensate for undesired Cannon and underactuated andEberhard flexible MBS is described by Seifried (2014); Seifried and (2009). In this case, the (1984)). Vibrations and elasticof deformations can aalso be to predicted based on an analysis the system. With modal Seifried and (2009). In this case, the ininto underactuated andEberhard flexible MBS is described by Seifried predicted based on anand analysis the system. With modal (1984)). Vibrations elasticof deformations can aalso be (2014); Seifried problem and Eberhard (2009). In this case, the inverse dynamics is stated as a two-point boundary predicted based on an analysis of the system. With ashaping modal (2014); analysis, Singer and Seering (1990) used input dynamics is stated as a two-point boundary (2014); Seifried problem and Eberhard (2009). In this case, the inanalysis, and Seering of (1990) used input predicted Singer based on an analysis the system. With ashaping modal verse verse dynamics problem is stated as a two-point boundary value problem initial and final conditions have analysis, Singer and Seering (1990)ofused input shaping methods cancel out the the at problem where where initial and as final conditionsboundary have to to verse dynamics problem is stated a two-point methods to canceland out Seering the vibrations vibrations the system system at some some value analysis, to Singer (1990)ofused input shaping value problem where initial and final conditions have to be met. The final condition, assures that the system ends methods to cancel out the vibrations of the system at some preselected frequencies. Staufer and Gattringer (2013) on be met. The final condition, assures that the system ends value problem where initial and final conditions have to preselected frequencies. Staufer and of Gattringer (2013) on be met. The final condition, assures that the system ends methods to cancel out the vibrations the system at some a stable manifold, i.e. a configuration space where all preselected frequencies. Stauferelement and Gattringer (2013) on on the other hand, used a lumped model model of the a stable i.e. a configuration space where all be met. Themanifold, final condition, assures that the system ends the other hand, used a lumped model model of the preselected frequencies. Stauferelement and Gattringer (2013) on on a stable manifold,toi.e. a configuration space wheregoes all trajectories aa given configuration as the other hand, usedsuitable a lumped element model model of the on system to compute control inputs. These feedback converge given configuration as time time on a stable converge manifold,toi.e. a configuration space wheregoes all system to hand, compute control inputs. These feedback the other usedsuitable a lumped element model model of the trajectories trajectories converge to a given configuration as time goes forward. On the other side, the initial condition forces system to compute suitable control inputs. These feedback and feedforward are complementary and can Onconverge the other initial condition trajectories to aside, giventhe configuration as timeforces goes and feedforward actions arecontrol complementary andfeedback can be be forward. system to computeactions suitable inputs. These Ontothe other side, the initial condition forces the system start on unstable manifold, i.e. conand feedforward actions are complementary anddone can by be forward. combined to improve robustness system start on an an unstable manifold, i.e. a aforces conforward. Ontothe other side, the initial condition combined to further further improve robustness as as was was and feedforward actions are complementary anddone can by be the systemspace to start onall antrajectories unstable manifold, i.e. agiven configuration where converge to a combined toLuca further improve robustness as was (2013). done by the for e.g., De (2000); Staufer and Gattringer where converge i.e. to aagiven the systemspace to start onall antrajectories unstable manifold, confor e.g., DetoLuca (2000); Staufer and Gattringer combined further improve robustness as was (2013). done by figuration figuration space where all trajectories converge to aisgiven configuration as time goes backward. This method able for e.g., De Lucaway (2000); Staufer and Gattringer (2013). An innovative to compute the feedforward action as where time goes backward. This method able figuration space all trajectories converge to aisgiven An innovative to compute the Gattringer feedforward action configuration for e.g., De Lucaway (2000); Staufer and (2013). configuration as time goes backward. This method isODE able to solve the inverse dynamics either in the form of an An innovative way to compute the feedforward action of flexible manipulators is presented in this work. The to solve the inverse dynamics either in the form of an ODE configuration as time goes backward. This method is able of manipulators is presented in this work.action The to solve the inverse dynamics either in the form of an ODE Anflexible innovative way to compute the feedforward in the form of a differential-algebraic equation (DAE), of flexible manipulators is presented in this work. The or or in the form of a differential-algebraic equation (DAE), to solve the inverse dynamics either in the form of an ODE of flexible manipulators is presented in this work. The or in the form ofuals differential-algebraic equation (DAE), publication is funded by the European Regional Development as shown by et to the The The publication is funded by the European Regional Development as shown by Br¨ Br¨ et al. al. (2014). (2014). In In order order to avoid avoid the or in the form ofuals differential-algebraic equation (DAE), Fund (ERDF) within European Union’s INTERREG A-program The publication is funded by the European Regional V Development as shown of by boundary Br¨ uls et al. (2014). In order to avoid the definition values, an optimization approach The(ERDF) Fund within European Union’s INTERREG V A-program publication is funded by the European Regional Development definition of boundary values, an optimization approach as shown by Br¨ u ls et al. (2014). In order to avoid the Greater Region,within project Robotix Union’s Academy. Fund (ERDF) European INTERREG V A-program definition of boundary values, an optimization approach was proposed by Bastos et al. (2013). This optimization Greater Region,within project Robotix Union’s Academy. Fund (ERDF) European INTERREG V A-program was proposed by Bastosvalues, et al. an (2013). This optimization definition of boundary optimization approach In addition, the project first author would like to acknowledge the Belgian Greater Region, Robotix Academy. was proposed by Bastos et al. (2013). This optimization method also to solution which converges In addition, the project first author would like to acknowledge the Belgian Greater Region, Robotix Academy. method also leads leads to aa bounded bounded solutionThis which converges was proposed by Bastos et al. (2013). optimization Fund for Research in Industry and Agriculture for its In addition, the first training author would like to acknowledge the Belgian method also leads to atwo-point bounded solution which converges Fund for Research in Industry and Agriculture for its to the solution of the boundary value problem In addition, the first training author would like to acknowledge the Belgian to the solution of the two-point boundary value problem method also leads to a bounded solution which converges financial (FRIA grant). Fund forsupport Research training in Industry and Agriculture for its financial (FRIA grant). to the solution of the two-point boundary value problem Fund forsupport Research training in Industry and Agriculture for its financial support (FRIA grant). to the solution of the two-point boundary value problem financial support (FRIA grant).
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when the length of the pre- and post-actuation phases increase (see Bastos et al. (2017)). Other numerical methods dealing with the inverse dynamics of flexible MBS in DAE form are compared by Moberg and Hanssen (2009). In the end, all methods lead to bounded and non-causal solutions. Here, the focus is given to the inverse dynamics of flexible 3D MBS for trajectory tracking tasks. The optimization formulation is chosen in combination with a finite element model of the system. Although the use of finite elements such as flexible beams, rigid bodies and kinematic joints elements can lead to a high dimensional model, it is a straight forward way to model general MBS, would it be a parallel or a serial system. A finite element method formulated on the special Euclidean group SE(3) is chosen. As shown by Br¨ uls et al. (2011), this formulation can benefit from a local frame formulation that reduces the nonlinearities of the equations of motion. Rotations in space can also be represented without singularity issues. In this paper, the use of finite elements in the SE(3) formalism is shortly described to understand how the flexible arm is modeled. Based on this model, the optimization problem, its objective function and its constraints are formulated. This constrained optimization problem is then solved using a direct transcription method (see Betts (2010)). Eventually, the case of a flexible serial manipulator is analyzed to illustrate this novel trajectory planning method. 2. MODEL OF A FLEXIBLE MULTIBODY SYSTEM 2.1 Finite element model The finite element method is a widespread tool when dealing with MBS. It is indeed a straight forward way to represent rigid bodies, flexible bodies and kinematic joints that compose a robotic arm. Here, the MBS is discretized in space into N nodes and a special Euclidean group SE(3) formalism is used to represent position and rotation. With such formalism, the position and orientation of each node I, with I = 1, ..., N , are represented as a homogeneous 4×4 matrix HI with a rotation RI ∈ SO(3) and a position pI ∈ R3 component. RI p I ∈ SE(3) HI = 0 1 The model is then constructed by connecting each node with rigid bodies, flexible bodies and kinematic joints (see Sonneville and Br¨ uls (2014); Sonneville et al. (2014)). In this work, flexible link are modeled using beam elements. Figure 1 gives an example of the discretization of a serial arm into N = 10 nodes connected with a rigid body, 4 beam elements and 3 kinematic joints. This example is further described and analyzed in the next section. Soft joints could also be considered by adding spring-damper parameters in the kinematic joints elements, but this is not considered here. 2.2 Equations of motion For trajectory planning and tracking problems, the endeffector position yef f of the flexible arm has to follow a prescribed path ypresc (t), evolving in time and defined by r scalar components. This can be achieved thanks to r
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Fig. 1. Discretization into N = 10 nodes of a flexible serial arm control inputs u = (u1 , ..., ur ). The dynamics of such MBS is governed by ˙ I = HI v ˜I (1) H Mv˙ + g(H, v) + BT λ = Au (2) Φ (H) = 0 (3) yef f (H) − ypresc (t) = 0 (4) In this set of equations, H = diag(H1 , ..., HN ) is the nodal configuration that gathers each nodal variable HI T T with I = 1, ..., N and v = (vIT , ..., vN ) is the vector of nodal velocities. The matrix M is the symmetric mass matrix of the system, g is the vector of internal and complementary inertia forces, B is the gradient of the m kinematic constraints Φ , which represent the connections imposed by the kinematic joints. Related to these kinematic constraints is the m dimensional vector λ of Lagrange multipliers. The matrix A is a boolean matrix applying the controls u to the system. The last equation is called the servo constraint (see Blajer and Kolodziejczyk (2004)) and compels the end-effector yef f to follow the prescribed trajectory ypresc (t). Flexible systems are said to be underactuated, in that case, the system has some internal dynamics of dimension 6N − m − r. With initial conditions on this internal dynamics, the behavior of the system would be completely defined. Unfortunately, the forward propagation of such initial conditions would lead to unbounded control inputs u requirements which are unachievable in reality. In order to find a bounded solution, no initial conditions of the system are defined and the inverse dynamics problem is formulated as an optimization problem. 3. OPTIMIZATION PROBLEM 3.1 Formulation As already mentioned previously, stable inversion techniques have to be used in order to have a bounded solution when solving the inverse dynamics of non-minimum phase systems. One possibility is to formulate the inverse problem as a constrained optimization problem. In our case, the objective is to minimize the internal dynamics of our flexible system. Considering that the internal dynamics can be represented using a function φ(H) depending on the nodal configurations, the optimization problem minimizes the objective function J on the given time lap T = tf − ti and can be mathematically written as tf 1 min J = min φ(H)dt (5) H H 2T t i This minimization has to be carried out by considering the constraints defined by the equations of motion of
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the flexible MBS given by (1)-(4) for t ∈ [ti , tf ]. In this formulation, no initial and final conditions on H and v are set and they are to be determined by the optimization process.
linked to each nodal configuration by (10) and involves the exponential mapping expSE(3) (˜•) that maps a 6×1 vector to an element of SE(3). k ) Hk+1 = Hk exp (∆Q (10) I
I
SE(3)
I
3.2 Direct transcription 3.4 Optimization variables To initialize the optimization, the initial guess of the variables, denoted with a hat ˆ •, are computed by solving the inverse dynamics of the equivalent rigid system. This rigid system can easily be solved using a computed torque method as, in this case, the system would have no internal dynamics. To solve the optimization problem, the direct transcription method is used and the evolution of the system is discretized into s time steps tk , with k = 1, ..., s, resulting in a discrete Nonlinear Program (NLP). Fig. 2 illustrates the direct transcription method for the configuration H.
After discretization, the unknown variables of the optimization problem are (H1 , v1 , v˙ 1 , a1 , λ 1 , u1 , ..., Hs , vs , v˙ s , as , λ s , us ) with Hk = diag(Hk1 , ..., HkN ) for k = 1, ..., s. The nodal variables HkI ∈ SE(3) belong to the SE(3) group and can therefore not be treated directly by classical NLP algorithms. To overcome this issue, an incremental configuration variable ∆q = (∆qT1 , ..., ∆qTN )T defined on a tangential space of SE(3) is introduced (see also Absil et al. (2008)). This new incremental variable determines ˆ and the current the variation between the initial guess H value H. At time step k, the relation between the configuration Hk = (Hk1 , ..., HkN ) and the incremental variable ∆qk = (∆qk1 , ..., ∆qkN ) is k ) ˆ k exp (11) (∆q Hk = H I
Fig. 2. Direct transcription method: optimization starting ˆ s ) that leads to the ˆ 1 , ..., H from the initial guess (H 1 s optimal trajectory (H , ..., H )
I
SE(3)
I
It is important to notice the difference between (10) and (11). The former relates the nodal configuration of the system at the same iteration but at two consecutive time steps, i.e. Hk and Hk+1 . The latter relates the current nodal configuration of the system to its initial guess at a ˆ k and Hk . Figure 3 is a zoom of Fig. 2 given time step, i.e. H and illustrates the relation between these variables where each arrow is an exponential mapping expSE(3) with either time incremental or iteration incremental arguments, i.e. ∆Qk and ∆qk respectively.
As the system is now discretized in time, the objective function written in its discrete form is now given by s 1 φ(Hk )h (6) J= 2T k=1
with h being the time step size. 3.3 Time integration scheme
In addition to the motion constraints given by (1)-(4), some time constraint must be considered in the discrete problem. The Lie group generalized-α integration scheme (see Br¨ uls et al. (2012)) is used here as it is able to deal with the equations of motion formulated in the Lie group SE(3). At each time step tk , these relations must be satisfied 1 ∆Qk − hvk − ( − β)h2 ak − βh2 ak+1 = 0 (7) 2 vk+1 − vk − (1 − γ)hak − γhak+1 = 0 (8) (1 − αm )ak+1 + αm ak − (1 − αf )v˙ k+1 − αf v˙ k = 0 (9)
where ak can be seen as an estimation of the acceleration, β, γ, αm and αf are parameters of the method, defining its behavior. The time incremental variable ∆Qk = k,T (∆Qk,T 1 , ..., ∆QN ) represents the incremental motion made by each node between two consecutive time steps tk and tk+1 . For each node I, this incremental variable is
Fig. 3. Relation between variables ∆qk , ∆Qk and Hk . In the end, the design variables x are x = (∆q1 , v1 , v˙ 1 , a1 , λ 1 , u1 , ..., ∆qs , vs , v˙ s , as , λ s , us ) and the optimization problem has only vectorial variables that can be treated using classical NLP solvers. The feedforward action u(t) can then be reconstructed at any time interpolation between the discrete values u1 , ..., us . 3.5 Optimization solver The combination of the finite element formalism and the direct transcription method leads to a large optimization problem with a high number of design variables and constraints. However, the problem is sparse as coupling
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terms are limited to consecutive time steps (see (7)-(10)) and nodes directly connected in the finite element mesh. Solvers such as KNITRO, IPOPT and FMINCON are able to deal with large scale and sparse problems. With appropriate options, the FMINCON solver in Matlab performed better for the tested cases and was chosen in the present work. To improve computation time, the gradients of the objective function J and the constraint equations are computed analytically and provided to the solver. 4. SERIAL MANIPULATOR EXAMPLE To illustrate the impact that can be done on the tracking precision by considering the flexible behavior of the MBS, let us consider the case of a flexible 3D serial manipulator. This manipulator has 3 actuated joints and two links as shown in Fig. 4. The first link has length l and a tubular square cross section of outer side a1 and edge thickness e1 . The second link also has length l but has a smaller tubular square cross section of side a2 and edge thickness e2 . The ratio between side a2 and thickness e2 of the second link is equal to the ratio a1 /e1 of the first link. In the initial configuration, each link is making a 45◦ angle with the horizontal plane and the input torques u1 ,u2 and u3 have their axis along z, y and y respectively. The point mass end-effector, with mass mend , is controlled using these three torque inputs. Flexibility is only considered in the second link and is modeled using 4 beam elements. The description of the beam formulation on SE(3) can be found in Sonneville et al. (2014).
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properties. A high numerical damping is considered for the optimization process; a spectral radius of ρ∞ = 0.01 is chosen (β = 0.98, γ = 1.48, αm = −0.97 and αf = 0.01). It is important to note that the convergence of the optimization process is quite sensitive to the choice of the initial guess. When gravity is acting on the system, the convergence is better when the initial guess starts with the static deflection of the arm. In this example, the integrand φ(Hk ) is defined as the strain energy of the flexible beams of the system. The number of time steps s is chosen in order to capture best the dynamics of the system: the inverse of the time step size 1/h is chosen to be five times higher than the first unstable frequency of the flexible system. For this serial arm, the first unstable frequency is 13 Hz and s = 150 steps is chosen, leading to 1/h = 100. This results in a optimization problem with around 39k variables that takes 5 iterations and around 2 minutes to solve (using a x64 bits i7-4600u CPU with 16 Gb RAM memory). For this case, Fig. 5 shows the input commands u and urigid , inputs with and without flexibility considerations respectively. Although some differences can be seen between these two control inputs, the pre-or post-actuation phases are not quite visible. By looking at the velocities of the joints, one can see in Fig. 6 that the velocity of joint 3 is still varying after the 1.3 s. Although the post-actuation torque is quite small, it still result in a motion of the joints of the system, i.e. internal motion of the system, during this post-actuation phase. Similar observations can be done for the pre-actuation phase. It is important to note that while this internal motion is present, the end-effector itself does not actually move during these pre- and post-actuation phases.
Commands [Nm]
10
Fig. 4. Serial 3D arm system with one rigid body and 4 beam elements.
Link 2 End-eff.
l=1m
a1 = 0.05 m
e1 = 0.01 m ρ = 2700 kg/m3
l=1m E = 70 GPa
a2 = 0.0075 m ν = 0.3
e2 = 0.0015 m ρ = 2700 kg/m3
u1 u2
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u3 u1,rigid u2,rigid u3,rigid 0
0.2
0.5
1
1.3
1.5
Time [s]
Fig. 5. Input commands of the flexible 3D arm. 2
Table 1. Parameters of the serial arm system. Link 1
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Joints velocity (rad/s)
The trajectory that has to be tracked is a planar circular arc of diameter d in the yz plane. The end-effector starts from position (2l cos(45◦ ), 0, 0) and goes to position (2l cos(45◦ ), d, 0). The motion profile is built using a seventh order polynomial to insure continuity of the positions, velocities, accelerations and jerks over time. The circular arc trajectory is covered in 1.1 s an the pre- and postactuation phases both last 0.2 s: the total simulation time is 1.5 s. Table 1 shows the material and geometrical parameters of the serial manipulator.
0
Joint 1 Joint 2 Joint 3 1
0
−1
mend = 0.1 kg −2
The initial guess is computed using a complete rigid model of the system. The second link is temporarily considered as a rigid body with the same geometrical and material
0
0.2
0.5
1
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Time (s)
Fig. 6. Joint velocities of the flexible 3D arm.
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Once the inverse dynamics is solved, the resulting optimized input torques u are applied to the system to perform a first direct dynamic analysis. In parallel, a second direct dynamic analysis is performed using the input torques urigid . These input commands result in an end-effector trajectory shown in Fig. 7. The relative tracking error resulting from both direct simulation can be calculated. For each time step tk , this relative error can be calculated using k k − yef ypresc f 2 (12) ek = k ypresc 2
where • 2 is the classical Euclidean norm or L2 norm and k = 1, ..., s. The relative rms error is calculated as s 1 (ek )2 (13) erms = s
Fig. 8. Evolution of the tracking error with respect to a2 . 5. CONCLUSION
k=1
Using the optimized inputs u, the relative rms error erms done on the trajectory drops down from 1.1% to 0.25%.
Y [m]
1
With u With urigid Prescribed
0
1.39
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1.4
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1.42
X [m]
Fig. 7. End-effector trajectory of the flexible 3D arm using u and urigid . For a given design of the serial manipulator, considering flexibility in the manipulator arm obviously result in a better tracking precision. This improvement can lead to a more lightweight design of the manipulator. We now consider a similar serial manipulator for which the side length a2 of the flexible second link is a parameter that will vary from amax = 0.1 m to amin = 0.0075 m. For each value of a2 , the control inputs that take into account flexibility and those where flexibility is not taken into account are both applied to the system. In order to be able to compare the tracking precision resulting from each inputs, an external perturbation force is added to the system’s end-effector during the direct dynamic analysis. For the results shown in Fig. 8, the perturbation is represented as an acceleration vector gp = (−0.3, −0.3, 0.3) m/s2 that has an equal magnitude along the x, y and z axes and that acts on the end-effector mass mend . It can be observed that for a relative tracking tolerance of 1.5%, the flexible control allows the design of a second link that has a side length of around 0.025 m instead of 0.05 m. For this particular example, the size of the second arm can be reduced by about half its size and still achieve similar performance as a rigidly controlled arm. The proposed feedforward control technique can thus be exploited to reduce the weight of the moving links and to increase their compliance while preserving the accuracy requirement.
Controlling the flexible behavior of lightweight arms can be challenging. Here, the improvement on the tracking precision coming from considering the flexible behavior of a manipulator are shown for a serial manipulator. A finite element model of the considered flexible MBS is first built using flexible beam elements, rigid bodies and kinematic joints. The inverse dynamics problem is then formulated as a constrained optimization problem where the objective is to minimize the internal dynamics of the flexible system. The use of a SE(3) formalism in the finite element model benefits from a local frame representation that reduces the nonlinearities of the constraint equations. The system is discretized in time and the direct transcription method is used to solve the optimization problem. The example of a serial link manipulator is discussed to show the improvement that can be done when flexibility is considered in the control scheme. By allowing more compliance in the manipulator, these adaptations would contribute to an increased safety of the machine as they induce a significant decrease in the severity of collisions in a human-robot cooperation application. In further works, a feedback loop could be added in the simulation to further analyze the impact of this feedforward command on the control of flexible manipulators. To reduce computation time, reduced order models based on a superelement formulation described by Sonneville (2015) can be considered to reduce the dimension of the optimization problem. REFERENCES Absil, P.A., Mahony, R., and Sepulchre, R. (2008). Optimization algorithms on Matrix Manifolds. Princeton University Press. Bastos, G.J., Seifried, R., and Br¨ uls, O. (2013). Inverse dynamics of serial and parallel underactuated multibody systems using a DAE optimal control approach. Multibody System Dynamics, 30, 359–376. doi: 10.1007/s11044-013-9361-z. Bastos, G.J., Seifried, R., and Br¨ uls, O. (2017). Analysis of stable model inversion methods for constrained underactuated mechanical systems. Mechanism and Machine Theory, 111, 99–117. Betts, J.T. (2010). Practical Method for Optimal Control and Estimation Using Nonlinear Programming. Advances in Desgin and Control. SIAM.
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Blajer, W. and Kolodziejczyk, K. (2004). A geometric approach to solving problems of control constraints: Theory and a dae framework. Multibody System Dynamics, 11, 343 – 364. Br¨ uls, O., Arnold, M., and Cardona, A. (2011). Two Lie group formulations for dynamic multibody systems with large rotations. In Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Washington D.C., U.S. Br¨ uls, O., Bastos, G.J., and Seifried, R. (2014). A stable inversion method for feedforward control of constrained flexible multibody systems. Journal of Computational and Nonlinear Dynamics, 9. doi:10.1115/1..4025476. Br¨ uls, O., Cardona, A., and Arnold, M. (2012). Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory, 48, 121–137. Cannon, R. and Schmitz, E. (1984). Initial experiments on the end-point control of a flexible one-link robot. The International Journal of Robotics Research, 3(3), 62–75. De Luca, A. (2000). Feedforward/feedback laws for the control of flexible robots. In Proceedings of the IEEE International Conference on Robotics & Automation. Devasia, S., Chen, D., and Paden, B. (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control, 41(7). Kwon, D.S. and Book, W.J. (1994). A time-domain inverse dynamic tracking control of a single link flexible manipulator. Journal of Dynamic Systems, Measurement and Control, 116, 193–200. Moberg, S. and Hanssen, S. (2009). Inverse dynamics of flexible manipulators. In Proceedings of the Multibody dynamics, ECCOMAS Thematic Conference. Seifried, R. (2014). Dynamics of Underactuated Multibody Systems: Modeling, Control and Optimal Design. Solid Mechanics and its Applications. Springer. Seifried, R. and Eberhard, P. (2009). Design of feedforward control for underactuated multibody systems with kinematic redundancy. Motion and Vibration Control: Selected papers from MOVIC 2008. Singer, N. and Seering, W.P. (1990). Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement and Control, 112, 76–82. doi: 10.1115/1.2894142. Sonneville, V. (2015). A geometric local frame approach for flexible multibody systems. Ph.D. thesis, University of Li`ege. Sonneville, V. and Br¨ uls, O. (2014). A formulation on the special Euclidean group for dynamic analysis of multibody systems. Journal of Computational and Nonlinear Dynamics, 9. doi:10.1115/1.4026569. Sonneville, V., Cardona, A., and Br¨ uls, O. (2014). Geometrically exact beam finite element formulated on the special euclidean group SE(3). Computer Methods in Applied Mechanics & Engineering, 268, 451–474. doi: 10.1016/j.cma.2013.10.008. Staufer, P. and Gattringer, H. (2013). Multibody System Dynamics, Robotic and Control, chapter PassivityBased Tracking Control of a Flexible Link Robot, 95– 112. Springer Vienna.
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