Page 1 Local Controllability Underactuated Manipulation Depends - PDF document

Northwestern University Lake Michigan Control Issues in Underactuated Robotic Manipulation Kevin M. Lynch Laboratory for Intelligent Mechanical Systems Department of Mechanical Engineering Northwestern University Evanston: main campus Evanston, IL USA Chicago: medical school, Workshop on Dynamics and Control law school Brussels, Belgium July 2002 Evanston LIMS Robotic Manipulation Laboratory for Intelligent Mechanical Systems The process of controlling the position (state) of one or more objects through contact forces by a robot. With Ed Colgate and Michael Peshkin Q: Where can a robot place a part? Research areas: Standard answer: Pick-and-place l Robotic manipulation —kinematic workspace, dexterous workspace l Motion planning for underactuated mechanical systems Other answers: Allow pushing, rolling, throwing, Human-robot inte rfaces l striking... l Haptic interfaces —dynamic workspace? Quasistatic Pushing Mechanics Motion Planning Allows placing parts by open-loop stable pushing Limit surface (Lynch and Mason 1996) determined by friction between object and support surface (Goyal, Ruina, Papadopoulos 1991) Page 1

Local Controllability Underactuated Manipulation Depends on: Underactuated robotic manipulation occurs when a robot controls more degrees-of-freedom of an object l Part geometry (or objects) than the robot has actuators. l Support friction (friction centroid) l Pushing friction Extra object DOF: rolling, slipping, flight coefficient Examples: —Robot assembly part velocities must —Parts feeders positively span a plane (not ω = 0) —Batting, juggling, pushing, rolling, throwing —Flexible objects Why? —Inexpensive, low-DOF robots Almost every part is locally controllable —Shift system complexity from hardware to motion planning and by pushing with a 2-DOF pushing point control Examples Related Work l Rolling Montana (1988) Li and Canny (1990) Dai and Brockett (1991) Bicchi and Sorrentino (1995) Hristu-Varsakelis (2001) Choudhury and Lynch (2002) l Juggling Buhler and Koditschek (1990) Rizzi and Koditschek (1993) 1 jo int r olling and throwing ar m Schaal and Atkeson (1993) Bishop and Spong (1999) Sony AP OS p arts feed er (a ut omatica lly planned, open loop ) Brogliato and Zavala-Rio (2000) Lynch and Black (2001) l Tapping Higuchi (1985) Huang and Mason (1998) l Pushing Mason (1986) Peshkin and Sanderson (1988) Alexander and Maddocks (1993) Lynch and Mason (1996) l Slipping Conveyor- based p arts feed er Trinkle (1992) Erdmann (1996) Plana r ju ggling Underactuated Manipulation Controllability z R = ( q R , dq R /dt ) ∈ M R = TC R Robot state: l Mechanics (nonprehensile manipulation) z P = ( q P , dq P /dt ) ∈ M P = TC P Part state: z = ( z R , z P ) ∈ M R × M P System state: Environment Hand controls ball Nonprehensile manipulation controls ball Underactuated manipulation: dim ( C R ) < dim ( C P ) —Pushing, rolling, slipping, throwing, batting —Friction, restitution, Newton s laws Given initial state z and time T , what is the set of —Object geometry, manipulator shape and motion reachable states R ( z,T )? constraints, unilateral constraints, changing dynamics (hybrid) l Controllability Part only: R P ( z,T ) —Reachability, feedability l Motion Planning l Feedback Control Page 2

Controllability (cont.) Controllability (cont.) l Accessible: R P ( z,†T ) is a full-dimensional subset l Accessible: R P ( z,†T ) is a full-dimensional subset of T z P M P for some T> 0. of T z P M P for some T> 0. l Feedable: z Pg ∈ R P ( z,†T ) for some T> 0 and any l Feedable: z Pg ∈ R P ( z,†T ) for some T> 0 and any z ∈ U, the set of initial possible states. z ∈ U, the set of initial possible states. l Controllable: z Pg ∈ R P ( z,T ) for some finite T and l Controllable: z Pg ∈ R P ( z,T ) for some finite T and any z, z Pg . any z, z Pg . l Locally controllable: z P ∈ int ( R P (z,†T )) for all z P l Locally controllable: z P ∈ int ( R P (z,†T )) for all z P and T> 0. (Only possible at zero velocity.) and T> 0. (Only possible at zero velocity.) l Equilibrium controllable: R P ( z,†T ) contains a l Equilibrium controllable: R P ( z,†T ) contains a neighborhood of q P at zero velocity. neighborhood of q P at zero velocity. Controllability (cont.) Single Input Systems l Accessible: R P ( z,†T ) is a full-dimensional subset l Minimum actuator systems of T z P M P for some T> 0. l Often globally controllable but not locally controllable l Feedable: z Pg ∈ R P ( z,†T ) for some T> 0 and any l Drift helps! z ∈ U, the set of initial possible states. l Controllable: z Pg ∈ R P ( z,T ) for some finite T and any z, z Pg . Planar juggler ( Bühler and Koditschek l Locally controllable: z P ∈ int ( R P (z,†T )) for all z P 1JOC conveyor parts feeder 1990; Zavala-Rio and Brogliato 1999; (Akella et al. 2000 ) and T> 0. (Only possible at zero velocity.) Lynch and Black 2001) l Equilibrium controllable: R P ( z,†T ) contains a Ball in an asymmetric bowl Planar body with one neighborhood of q P at zero velocity. (Choudhury and Lynch 2000) thruster (Lynch 1999) Single Input Systems (cont.) A Simple Model Robot shapes the natural dynamics of the environment. Repetitive throwing and A simple single input model: catching dz/dt = f ( z ) + g ( z ) u z system state f drift vector field (natural dynamics) g control vector field u control Butterfly (though often the systems are hybrid ) Page 3

Controllability Global Controllability dz/dt = f ( z ) + g ( z ) u, z ∈ M planar body, three thrusters planar body, two thrusters Involutive closure of { f, g } = Lie ({ f, g }) Theorem (Lian et al. 1994) If the drift vector field f linearly controllable locally controllable is Weakly Positively Poisson Stable (WPPS) and Lie ({ f, g }) = T z M ∀ z ∈ M , then the system is planar body, one thruster controllable. Accessibility + Poisson Stability ⇒ Controllability (Jurdjevic and Sussmann 1972; Lobry 1974; Brockett 1976; Bonnard 1981; not locally controllable (Lewis 1997, Manikonda and Jurdjevic 1997) Krishnaprasad 1997), but globally controllable (Lynch 1999) Poisson Stability Controllability Flow of drift field: Φ f : M × ℜ → M; ( z,t ) → Φ f ( z,t ) The point z is Positively Poisson Stable (PPS) for f if satellite attitude, two thrusters satellite attitude, one thruster for all T>0 and any neighborhood B ( z ) of z , there exists a time t>T such that Φ f ( z,t ) ∈ B ( z ) . f is PPS if the set of PPS points is dense in M . f is WPPS if for all z ∈ M , any neighborhood B ( z ) of z , and all T>0 , there exists t>T such that not locally controllable, but globally Φ f ( U z ,t ) ∩ B ( z ) ≠ ∅ . locally controllable controllable (Crouch 1984, Jurdjevic (Crouch 1984) 1997) Examples: a swing (no damping), satellite attitude, ball rolling in a bowl Extension Global Controllability If the drift is not WPPS, global controllability can be established by: controlled closed orbit Continuous fountain condition (Caines and Lemch 1999) Locally accessible states form an open subset of the state space. (Neither stronger nor weaker than local accessibility.) Plus some form of control recurrence , e.g., z = Φ ( z, T, u ) flow under the control u initial state for some control u and time T . Page 4

Global Controllability Motion Planning and Control A trajectory of a drift-free, controllable system is a nonsingular loop if 1) trajectory returns system to initial state, and controlled closed orbit 2) system is linearly controllable about the trajectory. (Sontag 1993; Sussmann 1993; Wen 1996) l Similar to the controlled closed orbit of systems with drift. accessible set l Generic loops are nonsingular for strongly (controllable about initial state closed orbit) accessible systems (Sontag 1993). Control Algorithm Summary Given: D is an open connected subset of M such that ∃ u r ( z ) ∈ int ( U ) ∀ z ∈ D. l Control parameterization u = ( u 1 , u 2 , , u k ) ∈ U l End-state map z 2 = f ( z 1 ,u ) Controllability on D l Goal state z g z ∈ int { f ( z,u ) | u ∈ B ( u r ( z )) } l Cost function V ( z ) (control Lyapunov function) B ( u r ( z )) is any neighborhood of u r ( z ) Stabilization of any point in D 1. Calculate recurrent control u r (z). l Define a distance between current and goal state. 2. u = u r - α ( ∂ V ( f ( z,u )) / ∂ u ) | u=u r , α > 0. l Perturb u r ( z ) to reduce the distance. 3. Execute control u . l Asymptotic stabilization if u r ( z ) gives nonsingular 4. Go to 1. loops. Example: Juggling Reversible States Point mass puck, zero thickness batter. Bold: reversible states z = ( x, y, x , y ) Circles: impact points One bat: z j+1 = f 1 ( z j ,u 1 ) u 1 = ( t 1 , ω 1 , t f ) (Buhler and Koditschek 1990; t = flight time, ω = impact vel Zavala-Rio and Brogliato 1999) Two bats: z j+2 = f 2 ( z j ,u 2 ) u 2 = ( t 1 , ω 1 , t 2 , ω 2 , t f ) D is the set of reversible states; puck can be batted (*) is cubic in flight time. At most three real solutions back and forth along the trajectory. to reversing impact states. Reversible impact states: x x + y y = 0 . (*) Page 5