Autonomous and Human-Robot Collaborative Manipulation of 1D and 2D Deformable Objects Without Modeling Deformation Dmitry Berenson Assistant Professor Robotics Engineering Computer Science
Manipulation of deformable objects • Deformable objects are ubiquitous in the home and medical settings • Present a method for manipulation deformable objects that does not require modeling or simulating deformation • Method is tested in simulation but has no knowledge of underlying models or simulation methods
Assumptions on Sensing • The geometry of the deformable object can be perceived Gripper • Though perception can be noisy Obstacle Deformable object • There are one or more grippers attached rigidly to the object Target points • Gripper configuration is known • The geometry of obstacles is known
Problem Statement • Compute a displacement 𝑟 𝑢 that moves current points of deformable object 𝑄 𝑢 as much as possible toward targets points 𝑈 : 𝑟 𝑢 𝑟 𝑢+1 𝑟 𝑢 Obstacle argmin 𝑒𝑗𝑡𝑢(𝑈, 𝑄 𝑢+1 ) 𝑟 𝑢 Deformable object points Target points 𝑈 𝑄 𝑢 𝑟 𝑢 should also compensate for excessive • stretching 𝑟 𝑢 should not bring the gripper(s) into • collision
Related Work: Modeling Deformable Objects • Can create models of deformable objects by probing [Lang et al., IJRR, 2002] [Cretu et al. ITIM, 2008] • Online modeling is an open problem
Related Work: Simulating Deformable Objects FEM simulation [Kaufmann et al., SIGGRAPH, 2008] Mass-spring model simulation [Desbrun et al., Graphics Interface, 1999] Mesh-less model simulation [Pezzementi et al., HIVETS, 2008] [Faure et al., SIGGRAPH, 2011] • Many methods are very sensitive to model discretization (e.g. FEM) • Accurate simulation requires significant computation time • Our approach avoids modeling and simulating deformation
Related Work: Motion Planning for Deformable Objects PRM for knot-tying Pre-computing deformation RRT in fully deformable environments [Saha et al., ISER, 2006] to enable fast planning [Rodriguez et al., ICRA, 2006] [Frank et al., IROS, 2011] • These methods all rely on access to a deformation simulator and accurate models
Related Work: Visual Servoing for Deformable Objects • Most methods require mesh-less or spring models Uses Jacobian of mesh-less Control law based on lattice-of-springs model reproducing kernel particle model [Hirai and Wada, Robotica, 2000] [Smolen and Patriciu, ACHI, 2009] [Wada et al., ICRA, 2001] • Adaptive Jacobian method requires initial guess for Jacobian • Number of interest points is limited Adaptive update to Jacobian [Navarro-Alarcon, ICRA, 2013]
Outline • Controller formulation • Computing the diminishing-rigidity Jacobian • Compensating for excessive stretching • Avoiding Collision • Simulation Results • Wrapping a string around a cylinder • Spreading a table cloth on a table • Human-robot Collaborative Cloth Folding
Using the Jacobian • Use Jacobian to compute displacement, given desired 𝑄 displacement of points 𝑄 = 𝐾 𝑟 𝑟 • Estimate the Jacobian 𝐾 𝑟 numerically requires simulation • Scales poorly with number of DOF • Computationally expensive • Potentially inaccurate
Heuristic for Approximating Deformation • Object exhibits diminishing rigidity near gripped points: • Parts of the object near gripped point move similarly to the gripper (i.e. as if rigidly attached) Parts that are farther away from the • gripper have smaller displacements • Not true in all cases, but method based on this is surprisingly effective
Approximating the Jacobian • Approximate 𝐾 𝑟 with the diminishing rigidity Jacobian 𝐾 𝑟 1. Write Jacobian of all points of the deformable object assuming they are rigidly attached to gripper(s) 2. Scale magnitude of each point’s row in the Jacobian proportional to distance from nearest gripper • For point 𝑗 , scaling is 𝑥 = 𝑓 −𝑙 𝐸(𝑗,𝑠𝑗𝑞𝑞𝑓𝑠) Geodesic distance along object • Then apply the pseudo-inverse to get 𝑟 : 𝐾 𝑟 + 𝑟 = 𝑄
Compensating for excessive stretching Assume we have knowledge of the object in its fully stretched-out state • • Pre-compute geodesics along object between each pair of points • Online, compute separation between each pair of points If a pair’s separation is approaching geodesic distance, move this pair closer together • Group all such movements in 𝑄 • s Combine with servoing movement: • 𝑟 = 𝐾 𝑟 + ( 𝑄 + 𝑄 s )
Prevent collisions of gripper with obstacles Combine “repulsion” of grippers from obstacles with servoing motion • • Smoothly vary magnitude of repulsion relative to magnitude of servoing • Allow servoing in null-space of collision avoidance 𝐾 𝑟 + ( 𝑄 + 𝑄 s ) For each gripper : • + 𝐾 𝑞 ′ = 𝛿 𝐾 𝑞 + 𝑦 𝑞 + I − 𝐾 𝑞 𝑟 𝑟 + (1 − 𝛿 ) 𝑟 𝛿 ∈ 0,1 : proportional to gripper g distance to nearest obstacle 𝐾 𝑞 : Jacobian of closest point 𝑞 of gripper g to nearest obstacle 𝑦 𝑞 : Movement of 𝑞 away from nearest obstacle
Results Demonstrated the controller on 3 example tasks in Bullet simulator • Winding string around cylinder Spreading a tablecloth Human-robot collaborative on a table cloth folding Method has no knowledge of underlying model or simulation method • used by simulator
Winding around cylinder 4x
String winding: Sensitivity to noise • Tested with Gaussian random noise injected into sensing of points of object Error • Method degrades gracefully with noise in this example
Spreading a tablecloth on a table 4x
Spreading tablecloth: Sensitivity to noise Tested with Gaussian random noise injected into sensing of points of • object Error Method can handle significant error, but diverges with 𝜏 ≥2.5cm (cloth • is 10mx10m)
Human-robot collaborative cloth folding • Controller matches its side of the object to user’s side • Targets for servoing determined by reflecting points on user’s side across plane of symmetry at each time step Autonomous User-controlled grippers grippers Plane of Symmetry
Human-robot collaborative cloth folding 4x
Compensating for excessive stretching 4x
Conclusions Present a method for manipulation deformable objects that does not require • modeling or simulating deformation • Method is tested in simulation but has no knowledge of underlying models or simulation methods • Future work Test on PR2 robot • Make Jacobian adaptive • • Test on 3D deformable objects
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