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Control problems for floating-base manipulators

Gianluca Antonelli, profile picture
Gianluca Antonelli

This document discusses control problems for floating-base manipulators. It begins with an introduction and definition of variables. It then covers inverse kinematics, possible kinematic solutions using behavioral control, possible dynamic solutions using virtual decomposition, and perspectives on simulations and experiments. The document outlines various tasks that could be controlled, such as end-effector position and orientation, joint limits, drag minimization, and obstacle avoidance. It also discusses using behavioral control approaches inspired by animal behavior to combine multiple tasks.

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Control problems for floating-base manipulators Gianluca Antonelli Universit`a di Cassino e del Lazio Meridionale antonelli@unicas.it http://webuser.unicas.it/lai/robotica http://www.eng.docente.unicas.it/gianluca antonelli Trondheim, 23 April 2014 Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Space, aerial and underwater vehicle-manipulators DLR Canadian Space Agency ALIVE normal robots but floating base kinematic coupling dynamic coupling unstructured environment Gianluca Antonelli Trondheim, 23 april 2014
Cooperation RedSea project RedSea project AMADEUS ARCAS Gianluca Antonelli Trondheim, 23 april 2014
Aerial Robotics Cooperative Assembly Gianluca Antonelli Trondheim, 23 april 2014
Aerial Robotics Cooperative Assembly ΣoΣe,1 Σe,2 Σv,1 Σv,2 Our task: cooperative control of the bar Gianluca Antonelli Trondheim, 23 april 2014
Marine Autonomous Robotics for InterventionS Gianluca Antonelli Trondheim, 23 april 2014
Marine Autonomous Robotics for InterventionS Our task: cooperative control of the bar (and logo design. . . ) Gianluca Antonelli Trondheim, 23 april 2014
Floating robots kinematics Oi η1 ηee ❅❅❘ end-effector velocities ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍❥ Jacobian system velocities˙ηee = ˙ηee1 ˙ηee2 = J(RI B, q)ζ ζ =   ν1 ν2 ˙q   Gianluca Antonelli Trondheim, 23 april 2014
UVMS dynamics in matrix form M(q)˙ζ + C(q, ζ)ζ + D(q, ζ)ζ + g(q, RI B) = τ formally equal to a ground-fixed industrial manipulator 1 however. . . Model knowledge Bandwidth of the sensor’s readings Vehicle hovering control Dynamic coupling between vehicle and manipulator External disturbances (current) Kinematic redundancy of the system 1 [Siciliano et al.(2009)Siciliano, Sciavicco, Villani, and Oriolo] [Fossen(2002)] [Schjølberg and Fossen(1994)] Gianluca Antonelli Trondheim, 23 april 2014
Dynamics Movement of vehicle and manipulator coupled movement of the vehicle carrying the manipulator law of conservation of momentum Need to coordinate at velocity level ⇒ kinematic control at torque level ⇒ dynamic control 2 2 [McLain et al.(1996b)McLain, Rock, and Lee] [McLain et al.(1996a)McLain, Rock, and Lee] Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -2- ˙σ = Jζ inverted by solving proper optimization problems Pseudoinverse ζ = J† ˙σ = JT JJT −1 ˙σ Transpose-based ζ = JT ˙σ Weighted pseudoinverse ζ = J† W ˙σ = W −1 JT JW −1 JT −1 ˙σ Damped Least-Squares ζ = JT JJT + λ2 Im −1 ˙σ need for closed-loop also. . . Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Redundancy may be used to add additional tasks ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Redundancy may be used to add additional tasks ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ ˙σa ✚✙ ✛✘ ˙σb ✙ ✖✕ ✗✔ ✶ Gianluca Antonelli Trondheim, 23 april 2014
Kinematic control scheme second. tasks ηd, qd τ η, q IK main task Control Output of kinematic control is the variable to be controlled by the actuators (vehicle thrusters and joints’ torques) Gianluca Antonelli Trondheim, 23 april 2014
A first kinematic solution Assuming the vehicle in hovering is not the best strategy to e.e. fine positioning3, better to kinematically compensate with the manipulator 3 [Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner] Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks Extended Jacobian4 Add additional (6 + n) − m constraints h(η, q) = 0 with associated Jh such that the problem is squared with ˙σ 0 = J Jh ζ 4 [Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker] Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks -2- Augmented Jacobian An additional task is given σh = h(η, q) with associated Jh such that the problem is squared with ˙σ ˙σh = J Jh ζ Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks -3- Task priority redundancy resolution σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + Jh I − J† J † ˙σh − JhJ† ˙σ Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks -4- Singularity robust task priority redundancy resolution 5 σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + I − J† J J† h ˙σh 5 we are talking about algorithmic singularities here. . . [Chiaverini(1997)] Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks -5- AMADEUS Agility task priority6 Task priority framework to handle both precision and set tasks Each task is the norm of the corresponding error (i.e., mi = 1) Recursive constrained least-squares within the set satisfying higher-priority tasks 6 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
Handling several tasks -6- Behavioral algorithms (behavior=task), bioinspired, artificial potentials, neuro-fuzzy, cognitive approaches, etc. btw. . . mood ? Gianluca Antonelli Trondheim, 23 april 2014
But. . . What are these tasks we are talking about ? Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Given 6 + n DOFs and m-dimensional tasks: End-effector position, m = 3 pos./orientation, m = 6 distance from a target, m = 1 alignment with the line of sight, m = 2 Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Manipulator joint-limits several approaches proposed, m = 1 to n, e.g. h(q) = n i=1 1 ci qi,max − qi,min (qi,max − qi)(qi − qi,min) Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Drag minimization, m = 1 7 h(q) = DT (q, ζ)W D(q, ζ) within a second order solution ˙ζ = J† ¨σ − ˙Jζ − k I − J† J ∂h ∂η ∂h ∂q + ∂h ∂ζ 7 [Sarkar and Podder(2001)] Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Manipulability/singularity, m = 1 h(q) = det JJT (In 8 priorities dynamically swapped between singularity and e.e.) joints inhibited direction singularity singularity setclose to 8 [Kim et al.(2002)Kim, Marani, Chung, and Yuh, Casalino and Turetta(2003)] [Chiacchio et al.(1991)Chiacchio, Chiaverini, Sciavicco, and Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Restoring moments: m = 3 keep close gravity-buoyancy of the overall system 9 m = 2 align gravity and buoyancy (SAUVIM is 4 tons) 10 fb fg τ 2 9 [Han and Chung(2008)] 10 [Marani et al.(2010)Marani, Choi, and Yuh] Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Obstacle avoidance m = 1 Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Workspace-related variables Vehicle distance from the bottom, m = 1 Vehicle distance from the target, m = 1 Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Sensors configuration variables Vehicle roll and pitch, m = 2 Misalignment between the camera optical axis and the target line of sight, m = 2 Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled Visual servoing variables Features in the image plane 11 11 [Mebarki et al.(2013)Mebarki, Lippiello, and Siciliano, Mebarki and Lippiello(in press, 2014)] Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
Competitive behavioral control Behaviors are in competitions and the higher priority can subsume the lower ones12 sensors behavior b ζ2 behavior a ζ1 behavior c ζ3 ζd 12 [Brooks(1986)] Gianluca Antonelli Trondheim, 23 april 2014
Cooperative behavioral control Behaviors cooperate and the priority is embedded in the gains13 sensors behavior b ζ2 α2 behavior a ζ1 supervisor α1 behavior c ζ3 α3 ζd 13 [Arkin(1989)] Gianluca Antonelli Trondheim, 23 april 2014
NSB Null Space-based Behavioral control Each action is decomposed in elementary behaviors/tasks motion reference command for each task ζd = J† ˙σd + Λσ σ = σd−σ Gianluca Antonelli Trondheim, 23 april 2014
NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + J† b ˙σb,d + Λbσb primary secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb ζd = ζa + Naζb Gianluca Antonelli Trondheim, 23 april 2014
NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
From behaviors to actions sensing/perception elementary behaviors actions commands supervisor Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: move to goal with obstacle avoidance obstacle avoidance σ1 = p − po ∈ R σ1,d = d J1 = ˆrT ∈ R1×2 ˆr = p − po p − po ζ1 = J† 1λ1 (d − p−po ) N(J1) = I − J† 1J1 = I − ˆrˆrT move to goal σ2 = p ∈ R2 σ2,d = pg J2 = I ∈ R2×2 ζ2 = Λ2 pg − p Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach ❆ ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach ❄ only obstacle avoidance Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach ❇ ❇ ❇❇◆ linear combination: higher task is corrupted Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB ❇ ❇ ❇◆ null-space-projection: higher task is fulfilled Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
Gain tuning Cooperative task a b c d situation 1 α1,1 α1,2 α1,3 α1,4 sit. 2 α2,1 α2,2 α2,3 α2,4 sit. 3 α3,1 α3,2 α3,3 α3,4 sit. 4 α4,1 α4,2 α4,3 α4,4 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
Stability analysis Lyapunov function14 V (˜σ) = 1 2 ˜σT ˜σ > 0 where ˜σ = ˜σT a ˜σT b ˜σT c T ˙V = −˜σT   Ja Jb Jc   v = −˜σT M ˜σ = −˜σT       Λa Oma,mb Oma,mc JbJ† aΛa JbNaJ† bΛb JbNJ† cΛc JcJ† aΛa JcNaJ† bΛb JcNJ† cΛc       ˜σ 14 [Antonelli(2009)] Gianluca Antonelli Trondheim, 23 april 2014
Stability results ˙V < 0 depending on the mutual relationships among the Jacobians: dependent ρ(J† x) + ρ(J† y) > ρ J† x J† y independent ρ(J† x) + ρ(J† y) = ρ J† x J† y orthogonal JxJ† y = Omx×my Gianluca Antonelli Trondheim, 23 april 2014
(Dis)advantages PROS Priorities strictly satisfied Analitical convergence results Easiest gain tuning Clear handling of the DOFs Competitive and cooperative as particular cases CONS Couples all the behaviors Impedance control not possible Gianluca Antonelli Trondheim, 23 april 2014
However. . . End effector going out of the workspace and one (eventually weighted) task always leads to singularity ❅ ❅ ❅❘ manipulator stretched Gianluca Antonelli Trondheim, 23 april 2014
Balance movement between vehicle and manipulator Need to distribute the motion e.g.: move mainly the manipulator when target in workspace move the vehicle when approaching the workspace boundaries move the vehicle for large displacement Some solutions, among them dynamic programming or fuzzy logic Gianluca Antonelli Trondheim, 23 april 2014
Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
Virtual Decomposition in a nutshell based on Newton-Euler formulation forward propagation of kinematic errors assuming 6 DOFs ˜ν1 ˜ν2 backward computation and projection of control forces/torques Gianluca Antonelli Trondheim, 23 april 2014
VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i Gianluca Antonelli Trondheim, 23 april 2014
VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i velocity propagation Gianluca Antonelli Trondheim, 23 april 2014
VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i joint contribution Gianluca Antonelli Trondheim, 23 april 2014
VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 Gianluca Antonelli Trondheim, 23 april 2014
VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 model-based compensation Gianluca Antonelli Trondheim, 23 april 2014
VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 feedback term Gianluca Antonelli Trondheim, 23 april 2014
VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 force propagation Gianluca Antonelli Trondheim, 23 april 2014
VD: cooperative control 1 Forward propagation until the object 2 Object control forces for the movement+internal 3 Forces projected to the link n of the two robots 4 Backward propagation hc,o hc,n+1 hc,n+1 Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Numerical simulation: underwater 6-DOF vehicle + 6-DOF manipulator Reach a pre-grasp configuration in terms of end-effector position and orientation initial configuration priority-1 task: e.e. configuration priority-2 task: vehicle roll+pitch priority-3 task: position of joint 2 only e.e. ⇒ complete solution ⇒ Gianluca Antonelli Trondheim, 23 april 2014
Numerical simulation: underwater 6-DOF vehicle + 7-DOF manipulator Cameraman action: keep the object in the field of view initial configuration priority-1 task: field of view priority-2 task: vehicle roll+pitch priority-3 task: mechanical joint limits animation ⇒ Gianluca Antonelli Trondheim, 23 april 2014
Numerical simulation: cooperative transportation of a bar The final objective for ARCAS kinematics-only no perception problems switch among actions Gianluca Antonelli Trondheim, 23 april 2014
First HW tests Tests made in Seville, November 2013 vehicle controller not implemented task: e.e. position+orientation simple e.e. hold no priority yet Gianluca Antonelli Trondheim, 23 april 2014
Additional HW tests Tests made in Seville, February 2014 vehicle controller to be tuned/improved vehicle obstacle avoidance e.e. + arm reconfiguration Gianluca Antonelli Trondheim, 23 april 2014
Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
Perspectives: the underactuated case Motivated by quadrotors control Roll and pitch functional to the horizontal movement ⇒ kinematic disturbances! pitch end effector Uncontrolled DOF affect the tasks Possible to handle in a task-priority approach? Gianluca Antonelli Trondheim, 23 april 2014
Perspectives: Null base reaction forces Dynamic model Mv M12 MT 12 Mq ˙ν ¨q + fc,v fc,q + fg,v fg,q = τv τq for the sole vehicle: Mv ˙ν + M12¨q + fc,v try to zeroing +fg,v = τv, achievable with: ¨q = −M† 12fc,v + N ¨qo yet another null space. . . Gianluca Antonelli Trondheim, 23 april 2014
Perspectives: arm’s motors not controllable in torque Several manipulators on the market equipped with position/velocity feedback control loops Need to design a proper controller for the sole vehicle Partially coupled model Iterative formulation Adaptive Simplest version: only gravity Mv ˙ν + M12¨q + fc,v + fg,v = τv, compensate only for fg,v h 0 0=Y (0)·θ0+U 0 1 Y (1) · θ1 + U 1 2h 2 2 +. . .= Y (0) U0 1Y (1) U0 1U1 2Y (2) . . . U0 1U1 2 . . . Un−1 n Y (n) Y         θ0 θ1 θ2 . . . θn         Gianluca Antonelli Trondheim, 23 april 2014
Perspectives: 2-arm-equipped vehicle Proposal to be submitted in few hours to H2020-ICT23 Fada-Catec Gianluca Antonelli Trondheim, 23 april 2014
Thanks for the attention The presented results are the outcome of the work of several colleagues from the University of Cassino, the Consortium ISME and PRISMA, the projects ARCAS and MARIS Filippo Arrichiello, Khelifa Baizid, Elisabetta Cataldi, Stefano Chiaverini, Amal Meddahi Gianluca Antonelli Trondheim, 23 april 2014
Bibliography I G. Antonelli. Stability analysis for prioritized closed-loop inverse kinematic algorithms for redundant robotic systems. IEEE Transactions on Robotics, 25(5):985–994, October 2009. G. Antonelli. Underwater robots. Springer Tracts in Advanced Robotics, Springer-Verlag, Heidelberg, D, 3rd edition, January 2014. G. Antonelli and S. Chiaverini. Fuzzy redundancy resolution and motion coordination for underwater vehicle-manipulator systems. IEEE Transactions on Fuzzy Systems, 11(1):109–120, 2003. R.C. Arkin. Motor schema based mobile robot navigation. The International Journal of Robotics Research, 8(4):92–112, 1989. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography II R.A. Brooks. A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, 2(1):14–23, 1986. G. Casalino and A. Turetta. Coordination and control of multiarm, nonholonomic mobile manipulators. In Proceedings IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 2203–2210, Las Vegas, NE, Oct. 2003. G. Casalino, E. Zereik, E. Simetti, S. Torelli A. Sperind`e, and A. Turetta. Agility for underwater floating manipulation: Task & subsystem priority based control strategy. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, PT, october 2012. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography III P. Chiacchio, S. Chiaverini, L. Sciavicco, and B. Siciliano. Closed-loop inverse kinematics schemes for constrained redundant manipulators with task space augmentation and task priority strategy. The International Journal Robotics Research, 10(4):410–425, 1991. S. Chiaverini. Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators. IEEE Transactions on Robotics and Automation, 13(3):398–410, 1997. S. Chiaverini, G. Oriolo, and I. D. Walker. Springer Handbook of Robotics, chapter Kinematically Redundant Manipulators, pages 245–268. B. Siciliano, O. Khatib, (Eds.), Springer-Verlag, Heidelberg, D, 2008. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography IV T.I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. J. Han and W.K. Chung. Coordinated motion control of underwater vehicle-manipulator system with minimizing restoring moments. In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, pages 3158–3163. IEEE, 2008. M. Hildebrandt, L. Christensen, J. Kerdels, J. Albiez, and F. Kirchner. Realtime motion compensation for ROV-based tele-operated underwater manipulators. In IEEE OCEANS 2009-Europe, pages 1–6, 2009. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography V J. Kim, G. Marani, WK Chung, and J. Yuh. Kinematic singularity avoidance for autonomous manipulation in underwater. Proceedings of PACOMS, 2002. G. Marani, S.K. Choi, and J. Yuh. Real-time center of buoyancy identification for optimal hovering in autonomous underwater intervention. Intelligent Service Robotics, 3(3):175–182, 2010. T.W. McLain, S.M. Rock, and M.J. Lee. Coordinated control of an underwater robotic system. In Video Proceedings of the 1996 IEEE International Conference on Robotics and Automation, pages 4606–4613, 1996a. T.W. McLain, S.M. Rock, and M.J. Lee. Experiments in the coordinated control of an underwater arm/vehicle system. Autonomous robots, 3(2):213–232, 1996b. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography VI R. Mebarki and V. Lippiello. Image-based control for aerial manipulation. Asian Journal of Control, in press, 2014. R. Mebarki, V. Lippiello, and B. Siciliano. Exploiting image moments for aerial manipulation control. In ASME Dynamic Systems and Control Conference, Palo Alto, CA, USA, 2013. N. Sarkar and T.K. Podder. Coordinated motion planning and control of autonomous underwater vehicle-manipulator systems subject to drag optimization. Oceanic Engineering, IEEE Journal of, 26(2):228–239, 2001. I. Schjølberg and T. Fossen. Modelling and control of underwater vehicle-manipulator systems. In in Proc. 3rd Conf. on Marine Craft maneuvering and control, pages 45–57, Southampton, UK, 1994. Gianluca Antonelli Trondheim, 23 april 2014
Bibliography VII B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo. Robotics: modelling, planning and control. Springer Verlag, 2009. Gianluca Antonelli Trondheim, 23 april 2014

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Control problems for floating-base manipulators

  • 1. Control problems for floating-base manipulators Gianluca Antonelli Universit`a di Cassino e del Lazio Meridionale antonelli@unicas.it http://webuser.unicas.it/lai/robotica http://www.eng.docente.unicas.it/gianluca antonelli Trondheim, 23 April 2014 Gianluca Antonelli Trondheim, 23 april 2014
  • 2. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 3. Space, aerial and underwater vehicle-manipulators DLR Canadian Space Agency ALIVE normal robots but floating base kinematic coupling dynamic coupling unstructured environment Gianluca Antonelli Trondheim, 23 april 2014
  • 5. Aerial Robotics Cooperative Assembly Gianluca Antonelli Trondheim, 23 april 2014
  • 6. Aerial Robotics Cooperative Assembly ΣoΣe,1 Σe,2 Σv,1 Σv,2 Our task: cooperative control of the bar Gianluca Antonelli Trondheim, 23 april 2014
  • 7. Marine Autonomous Robotics for InterventionS Gianluca Antonelli Trondheim, 23 april 2014
  • 8. Marine Autonomous Robotics for InterventionS Our task: cooperative control of the bar (and logo design. . . ) Gianluca Antonelli Trondheim, 23 april 2014
  • 9. Floating robots kinematics Oi η1 ηee ❅❅❘ end-effector velocities ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍❥ Jacobian system velocities˙ηee = ˙ηee1 ˙ηee2 = J(RI B, q)ζ ζ =   ν1 ν2 ˙q   Gianluca Antonelli Trondheim, 23 april 2014
  • 10. UVMS dynamics in matrix form M(q)˙ζ + C(q, ζ)ζ + D(q, ζ)ζ + g(q, RI B) = τ formally equal to a ground-fixed industrial manipulator 1 however. . . Model knowledge Bandwidth of the sensor’s readings Vehicle hovering control Dynamic coupling between vehicle and manipulator External disturbances (current) Kinematic redundancy of the system 1 [Siciliano et al.(2009)Siciliano, Sciavicco, Villani, and Oriolo] [Fossen(2002)] [Schjølberg and Fossen(1994)] Gianluca Antonelli Trondheim, 23 april 2014
  • 11. Dynamics Movement of vehicle and manipulator coupled movement of the vehicle carrying the manipulator law of conservation of momentum Need to coordinate at velocity level ⇒ kinematic control at torque level ⇒ dynamic control 2 2 [McLain et al.(1996b)McLain, Rock, and Lee] [McLain et al.(1996a)McLain, Rock, and Lee] Gianluca Antonelli Trondheim, 23 april 2014
  • 12. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 13. Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
  • 14. Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
  • 15. Kinematic control in pills -2- ˙σ = Jζ inverted by solving proper optimization problems Pseudoinverse ζ = J† ˙σ = JT JJT −1 ˙σ Transpose-based ζ = JT ˙σ Weighted pseudoinverse ζ = J† W ˙σ = W −1 JT JW −1 JT −1 ˙σ Damped Least-Squares ζ = JT JJT + λ2 Im −1 ˙σ need for closed-loop also. . . Gianluca Antonelli Trondheim, 23 april 2014
  • 16. Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Gianluca Antonelli Trondheim, 23 april 2014
  • 17. Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Redundancy may be used to add additional tasks ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Gianluca Antonelli Trondheim, 23 april 2014
  • 18. Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Redundancy may be used to add additional tasks ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ ˙σa ✚✙ ✛✘ ˙σb ✙ ✖✕ ✗✔ ✶ Gianluca Antonelli Trondheim, 23 april 2014
  • 19. Kinematic control scheme second. tasks ηd, qd τ η, q IK main task Control Output of kinematic control is the variable to be controlled by the actuators (vehicle thrusters and joints’ torques) Gianluca Antonelli Trondheim, 23 april 2014
  • 20. A first kinematic solution Assuming the vehicle in hovering is not the best strategy to e.e. fine positioning3, better to kinematically compensate with the manipulator 3 [Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner] Gianluca Antonelli Trondheim, 23 april 2014
  • 21. Handling several tasks Extended Jacobian4 Add additional (6 + n) − m constraints h(η, q) = 0 with associated Jh such that the problem is squared with ˙σ 0 = J Jh ζ 4 [Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker] Gianluca Antonelli Trondheim, 23 april 2014
  • 22. Handling several tasks -2- Augmented Jacobian An additional task is given σh = h(η, q) with associated Jh such that the problem is squared with ˙σ ˙σh = J Jh ζ Gianluca Antonelli Trondheim, 23 april 2014
  • 23. Handling several tasks -3- Task priority redundancy resolution σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + Jh I − J† J † ˙σh − JhJ† ˙σ Gianluca Antonelli Trondheim, 23 april 2014
  • 24. Handling several tasks -4- Singularity robust task priority redundancy resolution 5 σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + I − J† J J† h ˙σh 5 we are talking about algorithmic singularities here. . . [Chiaverini(1997)] Gianluca Antonelli Trondheim, 23 april 2014
  • 25. Handling several tasks -5- AMADEUS Agility task priority6 Task priority framework to handle both precision and set tasks Each task is the norm of the corresponding error (i.e., mi = 1) Recursive constrained least-squares within the set satisfying higher-priority tasks 6 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  • 26. Handling several tasks -6- Behavioral algorithms (behavior=task), bioinspired, artificial potentials, neuro-fuzzy, cognitive approaches, etc. btw. . . mood ? Gianluca Antonelli Trondheim, 23 april 2014
  • 27. But. . . What are these tasks we are talking about ? Gianluca Antonelli Trondheim, 23 april 2014
  • 28. Tasks to be controlled Given 6 + n DOFs and m-dimensional tasks: End-effector position, m = 3 pos./orientation, m = 6 distance from a target, m = 1 alignment with the line of sight, m = 2 Gianluca Antonelli Trondheim, 23 april 2014
  • 29. Tasks to be controlled Manipulator joint-limits several approaches proposed, m = 1 to n, e.g. h(q) = n i=1 1 ci qi,max − qi,min (qi,max − qi)(qi − qi,min) Gianluca Antonelli Trondheim, 23 april 2014
  • 30. Tasks to be controlled Drag minimization, m = 1 7 h(q) = DT (q, ζ)W D(q, ζ) within a second order solution ˙ζ = J† ¨σ − ˙Jζ − k I − J† J ∂h ∂η ∂h ∂q + ∂h ∂ζ 7 [Sarkar and Podder(2001)] Gianluca Antonelli Trondheim, 23 april 2014
  • 31. Tasks to be controlled Manipulability/singularity, m = 1 h(q) = det JJT (In 8 priorities dynamically swapped between singularity and e.e.) joints inhibited direction singularity singularity setclose to 8 [Kim et al.(2002)Kim, Marani, Chung, and Yuh, Casalino and Turetta(2003)] [Chiacchio et al.(1991)Chiacchio, Chiaverini, Sciavicco, and Gianluca Antonelli Trondheim, 23 april 2014
  • 32. Tasks to be controlled Restoring moments: m = 3 keep close gravity-buoyancy of the overall system 9 m = 2 align gravity and buoyancy (SAUVIM is 4 tons) 10 fb fg τ 2 9 [Han and Chung(2008)] 10 [Marani et al.(2010)Marani, Choi, and Yuh] Gianluca Antonelli Trondheim, 23 april 2014
  • 33. Tasks to be controlled Obstacle avoidance m = 1 Gianluca Antonelli Trondheim, 23 april 2014
  • 34. Tasks to be controlled Workspace-related variables Vehicle distance from the bottom, m = 1 Vehicle distance from the target, m = 1 Gianluca Antonelli Trondheim, 23 april 2014
  • 35. Tasks to be controlled Sensors configuration variables Vehicle roll and pitch, m = 2 Misalignment between the camera optical axis and the target line of sight, m = 2 Gianluca Antonelli Trondheim, 23 april 2014
  • 36. Tasks to be controlled Visual servoing variables Features in the image plane 11 11 [Mebarki et al.(2013)Mebarki, Lippiello, and Siciliano, Mebarki and Lippiello(in press, 2014)] Gianluca Antonelli Trondheim, 23 april 2014
  • 37. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 38. Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
  • 39. Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
  • 40. Competitive behavioral control Behaviors are in competitions and the higher priority can subsume the lower ones12 sensors behavior b ζ2 behavior a ζ1 behavior c ζ3 ζd 12 [Brooks(1986)] Gianluca Antonelli Trondheim, 23 april 2014
  • 41. Cooperative behavioral control Behaviors cooperate and the priority is embedded in the gains13 sensors behavior b ζ2 α2 behavior a ζ1 supervisor α1 behavior c ζ3 α3 ζd 13 [Arkin(1989)] Gianluca Antonelli Trondheim, 23 april 2014
  • 42. NSB Null Space-based Behavioral control Each action is decomposed in elementary behaviors/tasks motion reference command for each task ζd = J† ˙σd + Λσ σ = σd−σ Gianluca Antonelli Trondheim, 23 april 2014
  • 43. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + J† b ˙σb,d + Λbσb primary secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  • 44. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  • 45. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  • 46. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb ζd = ζa + Naζb Gianluca Antonelli Trondheim, 23 april 2014
  • 47. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  • 48. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  • 49. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  • 50. From behaviors to actions sensing/perception elementary behaviors actions commands supervisor Gianluca Antonelli Trondheim, 23 april 2014
  • 51. Simple comparison: move to goal with obstacle avoidance obstacle avoidance σ1 = p − po ∈ R σ1,d = d J1 = ˆrT ∈ R1×2 ˆr = p − po p − po ζ1 = J† 1λ1 (d − p−po ) N(J1) = I − J† 1J1 = I − ˆrˆrT move to goal σ2 = p ∈ R2 σ2,d = pg J2 = I ∈ R2×2 ζ2 = Λ2 pg − p Gianluca Antonelli Trondheim, 23 april 2014
  • 52. Simple comparison: competitive approach Gianluca Antonelli Trondheim, 23 april 2014
  • 53. Simple comparison: competitive approach ❆ ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 54. Simple comparison: competitive approach ❄ only obstacle avoidance Gianluca Antonelli Trondheim, 23 april 2014
  • 55. Simple comparison: competitive approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 56. Simple comparison: cooperative approach Gianluca Antonelli Trondheim, 23 april 2014
  • 57. Simple comparison: cooperative approach ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 58. Simple comparison: cooperative approach ❇ ❇ ❇❇◆ linear combination: higher task is corrupted Gianluca Antonelli Trondheim, 23 april 2014
  • 59. Simple comparison: cooperative approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 60. Simple comparison: NSB Gianluca Antonelli Trondheim, 23 april 2014
  • 61. Simple comparison: NSB ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 62. Simple comparison: NSB ❇ ❇ ❇◆ null-space-projection: higher task is fulfilled Gianluca Antonelli Trondheim, 23 april 2014
  • 63. Simple comparison: NSB ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  • 64. Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  • 65. Gain tuning Cooperative task a b c d situation 1 α1,1 α1,2 α1,3 α1,4 sit. 2 α2,1 α2,2 α2,3 α2,4 sit. 3 α3,1 α3,2 α3,3 α3,4 sit. 4 α4,1 α4,2 α4,3 α4,4 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  • 66. Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  • 67. Stability analysis Lyapunov function14 V (˜σ) = 1 2 ˜σT ˜σ > 0 where ˜σ = ˜σT a ˜σT b ˜σT c T ˙V = −˜σT   Ja Jb Jc   v = −˜σT M ˜σ = −˜σT       Λa Oma,mb Oma,mc JbJ† aΛa JbNaJ† bΛb JbNJ† cΛc JcJ† aΛa JcNaJ† bΛb JcNJ† cΛc       ˜σ 14 [Antonelli(2009)] Gianluca Antonelli Trondheim, 23 april 2014
  • 68. Stability results ˙V < 0 depending on the mutual relationships among the Jacobians: dependent ρ(J† x) + ρ(J† y) > ρ J† x J† y independent ρ(J† x) + ρ(J† y) = ρ J† x J† y orthogonal JxJ† y = Omx×my Gianluca Antonelli Trondheim, 23 april 2014
  • 69. (Dis)advantages PROS Priorities strictly satisfied Analitical convergence results Easiest gain tuning Clear handling of the DOFs Competitive and cooperative as particular cases CONS Couples all the behaviors Impedance control not possible Gianluca Antonelli Trondheim, 23 april 2014
  • 70. However. . . End effector going out of the workspace and one (eventually weighted) task always leads to singularity ❅ ❅ ❅❘ manipulator stretched Gianluca Antonelli Trondheim, 23 april 2014
  • 71. Balance movement between vehicle and manipulator Need to distribute the motion e.g.: move mainly the manipulator when target in workspace move the vehicle when approaching the workspace boundaries move the vehicle for large displacement Some solutions, among them dynamic programming or fuzzy logic Gianluca Antonelli Trondheim, 23 april 2014
  • 72. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  • 73. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  • 74. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  • 75. Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  • 76. Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  • 77. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 78. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  • 79. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  • 80. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  • 81. Virtual Decomposition in a nutshell based on Newton-Euler formulation forward propagation of kinematic errors assuming 6 DOFs ˜ν1 ˜ν2 backward computation and projection of control forces/torques Gianluca Antonelli Trondheim, 23 april 2014
  • 82. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i Gianluca Antonelli Trondheim, 23 april 2014
  • 83. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i velocity propagation Gianluca Antonelli Trondheim, 23 april 2014
  • 84. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i joint contribution Gianluca Antonelli Trondheim, 23 april 2014
  • 85. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 Gianluca Antonelli Trondheim, 23 april 2014
  • 86. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 model-based compensation Gianluca Antonelli Trondheim, 23 april 2014
  • 87. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 feedback term Gianluca Antonelli Trondheim, 23 april 2014
  • 88. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 force propagation Gianluca Antonelli Trondheim, 23 april 2014
  • 89. VD: cooperative control 1 Forward propagation until the object 2 Object control forces for the movement+internal 3 Forces projected to the link n of the two robots 4 Backward propagation hc,o hc,n+1 hc,n+1 Gianluca Antonelli Trondheim, 23 april 2014
  • 90. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 91. Numerical simulation: underwater 6-DOF vehicle + 6-DOF manipulator Reach a pre-grasp configuration in terms of end-effector position and orientation initial configuration priority-1 task: e.e. configuration priority-2 task: vehicle roll+pitch priority-3 task: position of joint 2 only e.e. ⇒ complete solution ⇒ Gianluca Antonelli Trondheim, 23 april 2014
  • 92. Numerical simulation: underwater 6-DOF vehicle + 7-DOF manipulator Cameraman action: keep the object in the field of view initial configuration priority-1 task: field of view priority-2 task: vehicle roll+pitch priority-3 task: mechanical joint limits animation ⇒ Gianluca Antonelli Trondheim, 23 april 2014
  • 93. Numerical simulation: cooperative transportation of a bar The final objective for ARCAS kinematics-only no perception problems switch among actions Gianluca Antonelli Trondheim, 23 april 2014
  • 94. First HW tests Tests made in Seville, November 2013 vehicle controller not implemented task: e.e. position+orientation simple e.e. hold no priority yet Gianluca Antonelli Trondheim, 23 april 2014
  • 95. Additional HW tests Tests made in Seville, February 2014 vehicle controller to be tuned/improved vehicle obstacle avoidance e.e. + arm reconfiguration Gianluca Antonelli Trondheim, 23 april 2014
  • 96. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  • 97. Perspectives: the underactuated case Motivated by quadrotors control Roll and pitch functional to the horizontal movement ⇒ kinematic disturbances! pitch end effector Uncontrolled DOF affect the tasks Possible to handle in a task-priority approach? Gianluca Antonelli Trondheim, 23 april 2014
  • 98. Perspectives: Null base reaction forces Dynamic model Mv M12 MT 12 Mq ˙ν ¨q + fc,v fc,q + fg,v fg,q = τv τq for the sole vehicle: Mv ˙ν + M12¨q + fc,v try to zeroing +fg,v = τv, achievable with: ¨q = −M† 12fc,v + N ¨qo yet another null space. . . Gianluca Antonelli Trondheim, 23 april 2014
  • 99. Perspectives: arm’s motors not controllable in torque Several manipulators on the market equipped with position/velocity feedback control loops Need to design a proper controller for the sole vehicle Partially coupled model Iterative formulation Adaptive Simplest version: only gravity Mv ˙ν + M12¨q + fc,v + fg,v = τv, compensate only for fg,v h 0 0=Y (0)·θ0+U 0 1 Y (1) · θ1 + U 1 2h 2 2 +. . .= Y (0) U0 1Y (1) U0 1U1 2Y (2) . . . U0 1U1 2 . . . Un−1 n Y (n) Y         θ0 θ1 θ2 . . . θn         Gianluca Antonelli Trondheim, 23 april 2014
  • 100. Perspectives: 2-arm-equipped vehicle Proposal to be submitted in few hours to H2020-ICT23 Fada-Catec Gianluca Antonelli Trondheim, 23 april 2014
  • 101. Thanks for the attention The presented results are the outcome of the work of several colleagues from the University of Cassino, the Consortium ISME and PRISMA, the projects ARCAS and MARIS Filippo Arrichiello, Khelifa Baizid, Elisabetta Cataldi, Stefano Chiaverini, Amal Meddahi Gianluca Antonelli Trondheim, 23 april 2014
  • 102. Bibliography I G. Antonelli. Stability analysis for prioritized closed-loop inverse kinematic algorithms for redundant robotic systems. IEEE Transactions on Robotics, 25(5):985–994, October 2009. G. Antonelli. Underwater robots. Springer Tracts in Advanced Robotics, Springer-Verlag, Heidelberg, D, 3rd edition, January 2014. G. Antonelli and S. Chiaverini. Fuzzy redundancy resolution and motion coordination for underwater vehicle-manipulator systems. IEEE Transactions on Fuzzy Systems, 11(1):109–120, 2003. R.C. Arkin. Motor schema based mobile robot navigation. The International Journal of Robotics Research, 8(4):92–112, 1989. Gianluca Antonelli Trondheim, 23 april 2014
  • 103. Bibliography II R.A. Brooks. A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, 2(1):14–23, 1986. G. Casalino and A. Turetta. Coordination and control of multiarm, nonholonomic mobile manipulators. In Proceedings IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 2203–2210, Las Vegas, NE, Oct. 2003. G. Casalino, E. Zereik, E. Simetti, S. Torelli A. Sperind`e, and A. Turetta. Agility for underwater floating manipulation: Task & subsystem priority based control strategy. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, PT, october 2012. Gianluca Antonelli Trondheim, 23 april 2014
  • 104. Bibliography III P. Chiacchio, S. Chiaverini, L. Sciavicco, and B. Siciliano. Closed-loop inverse kinematics schemes for constrained redundant manipulators with task space augmentation and task priority strategy. The International Journal Robotics Research, 10(4):410–425, 1991. S. Chiaverini. Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators. IEEE Transactions on Robotics and Automation, 13(3):398–410, 1997. S. Chiaverini, G. Oriolo, and I. D. Walker. Springer Handbook of Robotics, chapter Kinematically Redundant Manipulators, pages 245–268. B. Siciliano, O. Khatib, (Eds.), Springer-Verlag, Heidelberg, D, 2008. Gianluca Antonelli Trondheim, 23 april 2014
  • 105. Bibliography IV T.I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. J. Han and W.K. Chung. Coordinated motion control of underwater vehicle-manipulator system with minimizing restoring moments. In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, pages 3158–3163. IEEE, 2008. M. Hildebrandt, L. Christensen, J. Kerdels, J. Albiez, and F. Kirchner. Realtime motion compensation for ROV-based tele-operated underwater manipulators. In IEEE OCEANS 2009-Europe, pages 1–6, 2009. Gianluca Antonelli Trondheim, 23 april 2014
  • 106. Bibliography V J. Kim, G. Marani, WK Chung, and J. Yuh. Kinematic singularity avoidance for autonomous manipulation in underwater. Proceedings of PACOMS, 2002. G. Marani, S.K. Choi, and J. Yuh. Real-time center of buoyancy identification for optimal hovering in autonomous underwater intervention. Intelligent Service Robotics, 3(3):175–182, 2010. T.W. McLain, S.M. Rock, and M.J. Lee. Coordinated control of an underwater robotic system. In Video Proceedings of the 1996 IEEE International Conference on Robotics and Automation, pages 4606–4613, 1996a. T.W. McLain, S.M. Rock, and M.J. Lee. Experiments in the coordinated control of an underwater arm/vehicle system. Autonomous robots, 3(2):213–232, 1996b. Gianluca Antonelli Trondheim, 23 april 2014
  • 107. Bibliography VI R. Mebarki and V. Lippiello. Image-based control for aerial manipulation. Asian Journal of Control, in press, 2014. R. Mebarki, V. Lippiello, and B. Siciliano. Exploiting image moments for aerial manipulation control. In ASME Dynamic Systems and Control Conference, Palo Alto, CA, USA, 2013. N. Sarkar and T.K. Podder. Coordinated motion planning and control of autonomous underwater vehicle-manipulator systems subject to drag optimization. Oceanic Engineering, IEEE Journal of, 26(2):228–239, 2001. I. Schjølberg and T. Fossen. Modelling and control of underwater vehicle-manipulator systems. In in Proc. 3rd Conf. on Marine Craft maneuvering and control, pages 45–57, Southampton, UK, 1994. Gianluca Antonelli Trondheim, 23 april 2014
  • 108. Bibliography VII B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo. Robotics: modelling, planning and control. Springer Verlag, 2009. Gianluca Antonelli Trondheim, 23 april 2014