Control with signed (non negative) inputs stems from the unilaterality of the constraint and is present in sections 3 and 4. They may be signed impulsive inputs (in case of jugglers), or signed bounded inputs (in case of persistent contact) An interesting path is to investigate in which way the canonical dynamics in (32) could be used in the framework of persistent contact control of section 3. Also the problem of stabilization using controllers which switch between several kinds of sub-controllers (persistent contact, with impacts, PD+), could enlarge the scope of applications of the above results, see Remark 6. Finite-time convergent schemes (like sliding-mode control The time-discretization is also an important subject for contact tasks ,
Numerical Methods for Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics , Lecture Notes in Applied and Computational Mechanics, 2008. ,
URL : https://hal.archives-ouvertes.fr/inria-00423530
A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems: applications in electronics, Mathematical Programming, vol.167, issue.1, pp.31-67, 2011. ,
DOI : 10.1515/9781400873173
Analysis of collocated feedback controllers for four-bar planar mechanisms with joint clearances, Multibody System Dynamics, vol.32, issue.1, pp.101-136, 2016. ,
DOI : 10.1007/s11044-013-9356-9
URL : https://hal.archives-ouvertes.fr/hal-01218531
Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics, IEEE Transactions on Automatic Control, vol.59, issue.4, pp.876-891, 2014. ,
DOI : 10.1109/TAC.2014.2299335
URL : http://web.eecs.umich.edu/~grizzle/papers/CFL_Robotics_2012TAC.pdf
Modeling, control and simulation of upward jump of a biped, Multibody System Dynamics, vol.17, issue.7, pp.425-445, 2013. ,
DOI : 10.1109/ROBOT.1997.620036
URL : https://hal.archives-ouvertes.fr/hal-00718175
Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.359, issue.1789, pp.2327-2346, 1789. ,
DOI : 10.1098/rsta.2001.0854
A synchronization-based state observer for impact oscillators using only collision time information, International Journal of Robust and Nonlinear Control, vol.51, issue.1, pp.2542-2563, 2016. ,
DOI : 10.1016/j.sysconle.2003.09.007
URL : https://hal.archives-ouvertes.fr/hal-01326052
Theory, Facts, and Formulas with Application to Linear Systems Theory, Matrix Mathematics, 2005. ,
The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb???s friction: a tutorial, Multibody System Dynamics, vol.30, issue.2, pp.43-76, 2016. ,
DOI : 10.1007/s11044-013-9352-0
TRACKING CONTROL OF COMPLEMENTARITY LAGRANGIAN SYSTEMS, International Journal of Bifurcation and Chaos, vol.296, issue.06, pp.1839-1866, 2005. ,
DOI : 10.1109/JRA.1987.1087120
URL : https://hal.archives-ouvertes.fr/inria-00070619
Controllability in Linear Autonomous Systems with Positive Controllers, SIAM Journal on Control, vol.10, issue.2, pp.339-353, 1972. ,
DOI : 10.1137/0310026
Some perspectives on the analysis and control of complementarity systems, IEEE Transactions on Automatic Control, vol.48, issue.6, pp.918-935, 2003. ,
DOI : 10.1109/TAC.2003.812777
Absolute stability and the Lagrange???Dirichlet theorem with monotone multivalued mappings, Systems & Control Letters, vol.51, issue.5, pp.343-353, 2004. ,
DOI : 10.1016/j.sysconle.2003.09.007
URL : https://hal.archives-ouvertes.fr/inria-00071792
Inertial couplings between unilateral and bilateral holonomic constraints in frictionless Lagrangian systems, Multibody System Dynamics, vol.2, issue.6, pp.289-325, 2013. ,
DOI : 10.1007/978-3-642-01100-9
URL : https://hal.archives-ouvertes.fr/hal-00825581
Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction, Multibody System Dynamics, vol.8, issue.5, pp.175-216, 2014. ,
DOI : 10.1137/S1064827596297227
URL : https://hal.archives-ouvertes.fr/hal-00762986
Nonsmooth Mechanics. Models, Dynamics and Control, third edn. Communications and Control Engineering, 2016. ,
DOI : 10.1007/978-1-4471-0557-2
URL : https://hal.archives-ouvertes.fr/hal-01236953
Dissipative Systems Analysis and Control, Theory and Applications Communcations and Control Engineering, 2007. ,
DOI : 10.1007/978-1-84628-517-2
URL : https://hal.archives-ouvertes.fr/hal-01338369
On the controllability of linear juggling mechanical systems, Systems & Control Letters, vol.55, issue.4, pp.350-367, 2006. ,
DOI : 10.1016/j.sysconle.2005.08.012
URL : https://hal.archives-ouvertes.fr/inria-00071814
On the control of finite-dimensional mechanical systems with unilateral constraints, IEEE Transactions on Automatic Control, vol.42, issue.2, pp.200-215, 1997. ,
DOI : 10.1109/9.554400
On the control of complementary-slackness juggling mechanical systems, IEEE Transactions on Automatic Control, vol.45, issue.2, pp.235-246, 2000. ,
DOI : 10.1109/9.839946
Asymptotic stabilization of minimum phase nonlinear systems, IEEE Transactions on Automatic Control, vol.36, issue.10, pp.1122-1337, 1991. ,
DOI : 10.1109/9.90226
Popov???Belevitch???Hautus type controllability tests for linear complementarity systems, Systems & Control Letters, vol.56, issue.5, pp.381-387, 2007. ,
DOI : 10.1016/j.sysconle.2006.10.023
URL : http://iwi.eldoc.ub.rug.nl/FILES/root/2007/SysContLettCamlibel/2007SysContLettCamlibel.pdf
The Painlev?? paradox in contact mechanics, IMA Journal of Applied Mathematics, vol.81, issue.3, pp.538-588, 2016. ,
DOI : 10.1093/imamat/hxw027
On the Redundant-Drive Backlash-Free Robotic Mechanisms, Journal of Mechanical Design, vol.115, issue.2, pp.247-254, 1993. ,
DOI : 10.1115/1.2919184
Sliding Control Without Reaching Phase and Its Application to Bipedal Locomotion, Journal of Dynamic Systems, Measurement, and Control, vol.115, issue.3, pp.447-455, 1993. ,
DOI : 10.1115/1.2899122
URL : http://cyborg.seas.smu.edu/syslab/people/../papers/paper2.pdf
Dynamic model of ball bearings with internal clearance and waviness, Journal of Sound and Vibration, vol.294, issue.1-2, pp.23-48, 2006. ,
DOI : 10.1016/j.jsv.2005.10.005
Asymptotically Stable Running for a Five-Link, Four-Actuator, Planar Bipedal Robot, The International Journal of Robotics Research, vol.24, issue.6, pp.431-464, 2005. ,
DOI : 10.1109/9.664149
URL : https://hal.archives-ouvertes.fr/hal-00794892
Describing function of two masses with backlash, Nonlinear Dynamics, vol.27, issue.2???3, pp.409-413, 2009. ,
DOI : 10.1007/s11071-008-9410-y
Finite-Dimensional Inequalities and Complementarity Problems, Volume I. Springer Series in Operations Research, 2003. ,
Kinematics and Dynamics of Multibody Systems with Imperfect Joints, Lecture Notes in Applied and Computational Mechanics, vol.34, 2008. ,
Numerical and experimental investigation on multibody systems with revolute clearance joints, Nonlinear Dynamics, vol.24, issue.3, pp.383-398, 2011. ,
DOI : 10.1007/s11044-010-9220-0
URL : https://hal.archives-ouvertes.fr/hal-00649461
Trajectory tracking for a particle in elliptical billiards, International Journal of Control, vol.24, issue.2, pp.189-213, 2008. ,
DOI : 10.1103/PhysRevE.64.026212
Set-Valued Force Laws, Lecture Notes in Applied Mechanics, vol.1, 2001. ,
DOI : 10.1007/978-3-540-44479-4
An introduction to impacts Nonsmooth Mechanics of Solids, no, CISM Courses and Lectures, pp.45-102, 2006. ,
Asymptotically stable walking for biped robots: analysis via systems with impulse effects, IEEE Transactions on Automatic Control, vol.46, issue.1, pp.51-64, 2001. ,
DOI : 10.1109/9.898695
Comparative analysis of actuator concepts for active gear pair vibration control, Journal of Sound and Vibration, vol.269, issue.1-2, pp.273-294, 2004. ,
DOI : 10.1016/S0022-460X(03)00072-5
Fundamentals of Convex Analysis. Grundlehren text Editions, 2000. ,
Modeling, stability and control of biped robots???a general framework, Automatica, vol.40, issue.10, pp.1647-1664, 2004. ,
DOI : 10.1016/j.automatica.2004.01.031
URL : https://hal.archives-ouvertes.fr/inria-00072297
Vibration analysis of a rotating system due to the effect of ball bearing waviness, Journal of Sound and Vibration, vol.269, issue.3-5, pp.709-726, 2004. ,
DOI : 10.1016/S0022-460X(03)00127-5
Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches, Nonlinear Dynamics, vol.224, issue.6, pp.325-338, 2013. ,
DOI : 10.1243/13506501JET638
Non-smooth Behaviour of a Linkage Mechanism with Revolute Clearance Joints, New Advances in Mechanisms, Transmissions and Applications. Proceedings of the Second Conference MeTrApp 2013, Mechanisms and Machine Science, pp.233-241, 2014. ,
DOI : 10.1007/978-94-007-7485-8_29
Backlash Estimation With Application to Automotive Powertrains, IEEE Transactions on Control Systems Technology, vol.15, issue.3, pp.483-493, 2007. ,
DOI : 10.1109/TCST.2007.894643
URL : http://hem.hj.se/%7Elaad/Lagerberg_TCST_07.pdf
Modification of the Leuven integrated friction model structure, IEEE Transactions on Automatic Control, vol.47, issue.4, pp.683-687, 2002. ,
DOI : 10.1109/9.995050
The Theory of Matrices, 1985. ,
Stability and Convergence of Mechanical Systems with Unilateral Constraints, Lecture Notes in Applied and Computational Mechanics, vol.36, 2008. ,
DOI : 10.1007/978-3-540-76975-0
Simple mechanical control systems with constraints, IEEE Transactions on Automatic Control, vol.45, issue.8, pp.1420-1436, 2000. ,
DOI : 10.1109/9.871752
A novel transition model for lubricated revolute joints in planar multibody systems, Multibody System Dynamics, vol.9, issue.3, pp.279-294, 2016. ,
DOI : 10.1016/j.compstruc.2012.10.019
Hybrid Modelling and Control of Obstacle-Aided Snake Robot Locomotion, IEEE Transactions on Robotics, vol.26, issue.5, pp.781-799, 2010. ,
DOI : 10.1109/TRO.2010.2056211
Recurrence, controllability, and stabilization of juggling, IEEE Transactions on Robotics and Automation, vol.17, issue.2, pp.113-124, 2001. ,
DOI : 10.1109/70.928557
URL : http://lims.mech.nwu.edu/~lynch/papers/ieeetra2001/ieeetra2001.pdf
Analysis of Proportional-Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash, Journal of Vibration and Control, vol.9, issue.1, pp.119-156, 2003. ,
DOI : 10.1177/1077546303009001744
Feedback stabilization and tracking of constrained robots, IEEE Transactions on Automatic Control, vol.33, issue.5, pp.419-426, 1988. ,
DOI : 10.1109/9.1220
On the computation of the continuous-time reference trajectory for mechanical juggling systems, 2015 54th IEEE Conference on Decision and Control (CDC), pp.145-150, 2015. ,
DOI : 10.1109/CDC.2015.7402100
Velocity observers for non-linear mechanical systems subject to non-smooth impacts, Automatica, vol.38, issue.12, pp.2169-2175, 2002. ,
DOI : 10.1016/S0005-1098(02)00164-4
Control of (otherwise) uncontrollable linear mechanical systems through non-smooth impacts, Systems & Control Letters, vol.49, issue.4, pp.311-322, 2003. ,
DOI : 10.1016/S0167-6911(03)00101-4
Trajectory Tracking Control of Multiconstraint Complementarity Lagrangian Systems, IEEE Transactions on Automatic Control, vol.55, issue.6, pp.1300-1313, 2010. ,
DOI : 10.1109/TAC.2010.2042231
URL : https://hal.archives-ouvertes.fr/hal-00756307
Unilateral Contact and Dry Friction in Finite Freedom Dynamics, Nonsmooth Mechanics and Applications, no. 302 in CISM Courses and Lectures, pp.1-82, 1988. ,
DOI : 10.1007/978-3-7091-2624-0_1
Internal Preload Control of Redundantly Actuated Parallel Manipulators—Its Application to Backlash Avoiding Control, IEEE Transactions on Robotics, vol.21, issue.4, pp.668-677, 2005. ,
DOI : 10.1109/TRO.2004.842341
Multiple Impacts in Dissipative Granular Chains, Lecture Notes in Applied and Computational Mechanics, vol.72, p.36, 2014. ,
DOI : 10.1007/978-3-642-39298-6
URL : https://hal.archives-ouvertes.fr/hal-00908612
New models for backlash and gear play, International Journal of Adaptive Control and Signal Processing, vol.11, issue.1, pp.49-63, 1997. ,
DOI : 10.1002/(SICI)1099-1115(199702)11:1<49::AID-ACS394>3.0.CO;2-X
Controlling mechanical systems with backlash???a survey, Automatica, vol.38, issue.10, pp.1633-1649, 2002. ,
DOI : 10.1016/S0005-1098(02)00047-X
Finite time stabilization of a perturbed double integrator with unilateral constraints, Removal of singularities in the inverse dynamics of parallel robots. Mechanism and Machine Theory, pp.200-212, 2014. ,
DOI : 10.1016/j.matcom.2012.02.011
Globally asymptotically stable ???PD+??? controller for robot manipulators, International Journal of Control, vol.6, issue.6, pp.1697-1712, 1988. ,
DOI : 10.1115/1.3139651
Vibro-Impact Problems with Dry Friction---Part I: Existence Result, SIAM Journal on Mathematical Analysis, vol.47, issue.5, pp.3285-3313, 2015. ,
DOI : 10.1137/140988899
Vibro-Impact Problems with Dry Friction---Part II: Tangential Contacts and Frictional Catastrophes, SIAM Journal on Mathematical Analysis, vol.48, issue.2, pp.1272-1296, 2016. ,
DOI : 10.1137/140988905
Penalty approximation for dynamical systems submitted to multiple non-smooth constraints, Multibody System Dynamics, vol.8, issue.3, pp.347-366, 2002. ,
DOI : 10.1006/jdeq.2001.4027
URL : https://doi.org/10.1006/jdeq.2001.4027
A critical overview of internal and external cylinder contact force models, Nonlinear Dynamics, vol.220, issue.3, pp.681-697, 2011. ,
DOI : 10.1023/B:MUBO.0000049131.77305.d8
URL : https://hal.archives-ouvertes.fr/hal-00624469
Applicability domain of internal cylindrical contact force models. Mechanism and Machine Theory 78, pp.141-157, 2014. ,
DOI : 10.1016/j.mechmachtheory.2014.03.010
Multibody Dynamics with Unilateral Contacts. Nonlinear Science, 1996. ,
Stability Analysis and Control of Rigid-Body Systems With Impacts and Friction, IEEE Transactions on Automatic Control, vol.61, issue.6, pp.1423-1437, 2016. ,
DOI : 10.1109/TAC.2015.2459151
Dynamics and control of a class of underactuated mechanical systems, IEEE Transactions on Automatic Control, vol.44, issue.9, pp.1663-1671, 1999. ,
DOI : 10.1109/9.788533
Contribution to the modeling of nonsmooth multipoint contact dynamics of biped locomotion?theory and experiments, Robotica, vol.26, pp.157-175, 2007. ,
URL : https://hal.archives-ouvertes.fr/hal-00919720
Global Controllability of Linear Systems with Positive Controls, SIAM Journal on Control, vol.11, issue.3, pp.417-423, 1973. ,
DOI : 10.1137/0311034
Controllability of Linear Oscillatory Systems Using Positive Controls, SIAM Journal on Control, vol.9, issue.2, pp.253-262, 1971. ,
DOI : 10.1137/0309019
An Optimal Trajectory Planner for a Robotic Batting Task: The Table Tennis Example, Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics, pp.90-101, 2016. ,
DOI : 10.5220/0005982000900101
Comments on "Adaptive manipulator control: a case study" by J. Slotine and W. Li, IEEE Transactions on Automatic Control, vol.35, issue.6, pp.761-762, 1990. ,
DOI : 10.1109/9.53565
Observer Design for Unilaterally Constrained Lagrangian Systems: A Passivity-Based Approach, IEEE Transactions on Automatic Control, vol.61, issue.9, pp.2386-2401, 2016. ,
DOI : 10.1109/TAC.2015.2492098
URL : https://hal.archives-ouvertes.fr/hal-01113344
Introduction to modelling and parameter identification methodology of linkages by measurements and simulation, Proceedings of 13rd World Congress in Mechanisms and Machine Science, pp.19-25, 2011. ,
Measurements and simulations of a crank and rocker mechanism including friction, clearance and impacts, Proceedings of IXth Intern. Conf. on the Theory of Machines and Mechanisms, 2004. ,
Optimal dynamic design of a planar slider-crank mechanism with a joint clearance, Mechanism and Machine Theory, vol.86, pp.191-200, 2015. ,
DOI : 10.1016/j.mechmachtheory.2014.12.008
Hybrid zero dynamics of planar biped walkers, IEEE Transactions on Automatic Control, vol.48, issue.1, pp.42-56, 2003. ,
DOI : 10.1109/TAC.2002.806653
Online adaptation of reference trajectories for the control of walking systems, Robotics and Autonomous Systems, vol.54, issue.7, pp.559-566, 2006. ,
DOI : 10.1016/j.robot.2006.04.007
URL : https://hal.archives-ouvertes.fr/inria-00070702
Modeling of static friction in closed-loop kinematic chains???Uniqueness and parametric sensitivity problems, Multibody System Dynamics, vol.83, issue.4, pp.10-1007, 2016. ,
DOI : 10.1007/s11071-015-2485-3
Attractivity of Equilibrium Sets of Systems with Dry Friction, Nonlinear Dynamics, vol.35, issue.1, pp.19-39, 2008. ,
DOI : 10.1023/B:NODY.0000017482.61599.86
On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface, European Journal of Mechanics - A/Solids, vol.49, pp.315-328, 2015. ,
DOI : 10.1016/j.euromechsol.2014.08.005
A comprehensive model for 3D revolute joints with clearances in mechanical systems, Nonlinear Dynamics, vol.24, issue.1, pp.309-328, 2015. ,
DOI : 10.1007/s11044-010-9192-0
On the control of a one degree-of-freedom juggling robot, Dynamics and Control, vol.9, issue.1, pp.67-90, 1999. ,
DOI : 10.1023/A:1008346825330
Direct adaptive control design for one-degree-of-freedom complementary-slackness jugglers, Automatica, vol.37, issue.7, pp.1117-1123, 2001. ,
DOI : 10.1016/S0005-1098(01)00061-9