Omnidirectional Vision-Based Formation Control René Vidal Omid Shakernia Shankar Sastry Department of EECS, UC Berkeley

Motivation Formation control is ubiquitous Safety in numbers Decreased aerodynamic drag Higher traffic throughput Applications in defense, space exploration, etc Formation control is a problem that has received a lot of attention in the control/robotics community over the past few years. It is a problem that is ubiquitous both in nature and in man made environments. On the top left, we have a duck being followed by baby ducks in a straight line: “String formation” On the bottom left we have a school of fish swimming together to reduce the probability of being eaten by a bigger fish (safety in numbers) On the middle left, we have geese flying in a V-Formation. This formation is actually quite interesting: A geese in the formation can fly 71% further than a geese outside the formation, due to reductions In the aerodynamic drag. On the right, we have man made applications in Defense: the commander can fly in the middle and be protected by the rest of the team Automatic highways: a platoon of cars can increase traffic throughput, while maintaining safety Space exploration: a cluster of satellites with telescopes can better explore the space by moving In a formation Top left duck duckling, line formation School of fish Goose, Geese flying in a V-formation: reduces the drag can fly 71% further than a geese outside the formation They fly on top of the vortex : when the guy in front moves the flags, It forms a vortex of air raising up that cushiones safety in numbers means for example, if a fish is swimming in a large school of other fish he is less likely to get eaten than if he is swimming alone Commander protection in the middle. Explore space better by having a cluster with many satelites with telescopes are better for space exploration if they move in a formation

Previous Work Formation control has a very rich literature (short list): String stability for line formations [Swaroop et.al. TAC96] Formation can become unstable due to error propagation Mesh stability of UAVs [Pant et.al. ACC01] Generalization of string stability to a planar mesh Input-to-state stability [Tanner et.al ICRA02] Structure of interconnections and amount of information communicated affects ISS Feasible formations [Tabuada et.al. ACC01] Differential geometric conditions on feasibility of formations under kinematic constraints of mobile robots Vision-based formation control [Das et.al. TAC02] Leader position estimated by vision; Formation control in task space

Our Approach Distributed formation control (no explicit communication) Formation specified in image plane of each follower Multi-body motion segmentation to estimate leader position Followers employ tracking controller in the image plane Naturally incorporate collision avoidance by exploiting geometry of omni-directional images

Outline Omnidirectional vision Distributed formation control Catadioptric cameras, back-projection ray Paracatadioptric optical flow equations Multi-body motion segmentation Distributed formation control Leader-follower dynamics in image plane Feedback linearization control design Collision avoidance using navigation functions Experimental results Motion segmentation of robots in real sequence Vision-based formation control simulations

Catadioptric Camera Catadioptric camera is lens-mirror combination Single effective focal point Parabolic mirror, orthographic lens Hyperbolic camera, perspective lens Efficiently compute back-projection ray associated with each pixel in image

Paracatadioptric Projection Model Paraboloidal mirror surface: Projection model: Back-projection ray for pixel Paracatadioptric Projection Perspective Projection

Paracatadioptric Optical Flow Optical flow induced by camera motion of angular velocity and linear velocity Camera motion restricted to X-Y plane Paracatadioptric Optical Flow Perspective Optical Flow

Multi-body Motion Segmentation Optical flows of pixels in frames live in a 5 dimensional subspace. Optical flows can be factorized into structure and motion Given independent motions Number of independent motions is obtained as:

Motion Segmentation Independent motions lie in orthogonal subspaces Block diagonal structure of W implies leading singular vector will have the same value for pixels corresponding to same rigid motion Eigenvector has different value for pixels corresponding to different motions The largest group of pixels (static background) is flow induced by camera motion Roots of a polynomial

Leader-Follower Dynamics Kinematic model Inputs Leader position in follower’s camera Write as drift-free control system: Recover leader velocity using optical flow of background Paracatadioptric leader-follower dynamics

Cartesian coordinates Control in Image Plane Cartesian coordinates Polar coordinates Controlling in Cartesian coordinates, leader trajectory intersects circle Controlling in polar coordinates, follower mostly rotates Trajectory passing through inner circle is a collision

Feedback Linearization Control Law in Polar Coordinates Leader position , desired leader position Degenerate configurations due to nonholonomy of mobile robot Robot can not move sideways instantaneously due to geometry of catadioptric projection Corresponds to horizon points at infinity Feedback Linearization Control Law in Polar Coordinates

Distributed Formation Controller Enhanced collision avoidance using a Navigation Function Control law incorporating collision avoidance

Experimental Results Multi-body motion segmentation

Wedge Formation Green follows red Blue follows red

String Formation Green follows red Blue follows green

Conclusions A framework for distributed formation control in the omni-directional image plane An algorithm for multi-body motion segmentation in omni-directional images Future work Generalize formation control to UAV dynamics Hybrid theoretic formation switching control Implement on BEAR fleet of UGVs and UAVs