Cooperative Path Following of Robotic Vehicles using Event based - PowerPoint PPT Presentation

Cooperative Path Following of Robotic Vehicles using Event based Control and Communication R. Praveen Jain, A. Pedro Aguiar, Jo˜ ao Borges de Sousa Department of Electrical and Computer Engineering Faculdade de Engenharia, Universidade do Porto Porto, Portugal January 17, 2017 This work was supported by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642153.

Outline of Presentation Introduction Path Following Control Design Event-based Cooperative Control Event-based Control (Consensus) Event-based Communication Experiment Results Open Problems

Introduction Continuous inter-robot communication! 1. Practical? Considering... ◮ Communication hardware. ◮ Bandwidth and Power 2. Necessary? ◮ Need for methods that reduce frequency of communication be- tween the robots! ◮ Event-triggered Consensus and Self-triggered Consensus algo- rithms applied to the Cooperative Path Following (CPF) prob- lem.

Cooperative Path Following Framework Virtual Point ◮ A two stage control architecture. ◮ Lower layer: Path Following (PF) controller. 1. Responsible for motion control of individual robot. 2. Follows a pre-specified geometric path (no temporal constraints)

Cooperative Path Following Framework Aligned! Requires Consensus!! ◮ Higher layer: Cooperative Controller (CC) 1. Responsible for cooperation among multiple robots. 2. First order Consensus controller. 3. Main results: Self-triggered approach 1 and Event-triggered approach 2 1 Jain, R. Praveen, A. Pedro Aguiar, and Jo˜ ao Borges de Sousa. ”Self-triggered cooperative path following control of fixed wing Unmanned Aerial Vehicles.” In International Conference on Unmanned Aircraft Systems (ICUAS), pp. 1231-1240. IEEE, 2017. 2 Jain, R. Praveen, A. Pedro Aguiar, and Jo˜ ao Borges de Sousa. ”Cooperative Path Following of Robotic Vehicles using an Event based Control and Communication Strategy.” Accepted to the International Conference on Robotics and Automation (ICRA), 2018.

Path Following Control Design

System Model Assumptions 1. 2D operation, extension to 3D case straight forward. 2. Inner loop controller able to track the reference control com- mands generated by the PF controller. System Model ˙ p i ( t ) = R i ( t ) v i ( t ) + w i ( t ) ˙ R i ( t ) = R i ( t ) S ( ω i ) where p i ∈ R 2 - position of the robot w.r.t inertial frame { I } , v i ∈ R 2 = [ v f i 0] T - linear velocity of the robot w.r.t body frame { B } , R i ∈ SO (2), S ( ω i ( t )) ∈ so (2) and ω i ∈ R is input angular velocity, u i ( t ) = [ v f i ω i ] T - control inputs for the vehicle,

Problem Formulation ◮ Consider a given reference geometric Virtual Point path p d i ( γ i ) : R → R 2 parameterized by the path variable γ i ∈ R . ◮ A desired speed assignment v d ∈ R . Control Objective ◮ Design u i ( t ) such that the path following error, � p i − p d i ( γ i ) � converges to an arbitrary small neighborhood of the origin as t → ∞ . ◮ The desired speed assignment, � ˙ γ i − v d � → 0 as t → ∞ .

Error Dynamics ◮ Define error variable e i = R T i ( p i − p d i ( γ i )) + ǫ ◮ The error dynamics satisfies ∂ p d i ( γ i ) e i = − S ( ω i ) e i + ∆ u i − R T ˙ γ i ˙ i ∂γ i � 1 � − ǫ 2 , u i = [ v f i ω i ] T and ǫ = [ ǫ 1 ǫ 2 ] T � = 0. where ∆ = 0 ǫ 1 ◮ Impose v i γ i = v d + ˜ ˙ r + g i ( t ) v i where ˜ r is additional control input used for achieving cooper- ation and g i ( t ) is the path following error correction term with � g i ( t ) � ≤ µ A. Alessandretti, A. P. Aguiar and C. N. Jones, ”Trajectory-tracking and path-following controllers for constrained underactuated vehicles using Model Predictive Control,” 2013 European Control Conference (ECC), Zurich, 2013, pp. 1371-1376.

Control Law Theorem: Path Following Controller Given the error dynamics for the path following system, the estimate of error states ˆ e i ( t ) = e i ( t ) + ˜ e i ( t ), the control law � ∂ p d i ( γ i ) � u i = ∆ − 1 e i + R T − K p ˆ v d i ∂γ i makes the closed-loop system Input-to-State Stable (ISS) with re- spect to the estimation error ˜ e i ( t ), the formation speed actuation v i signal ˜ r ( t ) and path following error correction term g i ( t ).

Event based Cooperative Control Control and Communication

Problem Formulation ◮ Consider N robots with associated reference path p d i ( γ i ) pa- rameterized by γ i for i = 1 , 2 , · · · , N . ◮ Let ˙ v i γ i = v d + ˜ r + g i . Control Objective v i Design decentralized, event-triggered control law for ˜ r such that, 1. � γ i − γ j � → 0 for all i , j = 1 , · · · , N and i � = j as t → ∞ . 2. Each robot communicates and updates control action at event time instants t i k determined by an Event Triggering Condition

First Order Consensus ◮ Consider N agents modeled as single integrator dynamics γ i = u i ( t ) ˙ ◮ Known result on continuous time average consensus for undi- rected graphs: � u i ( t ) = − γ i ( t ) − γ j ( t ) = − L γ ( t ) j ∈N i where L is the graph Laplacian Controller is implemented continuously! Neighbor states are measured continuously!

Step 1: Event-triggered Consensus Theorem: Event-triggered Consensus The decentralized, event-triggered consensus controller � ( γ i ( t i k ) − γ j ( t i k )) = [ L γ ( t i u i ( t ) = − k )] i j ∈N i defined over t ∈ � k ∈ Z ≥ 0 [ t i k , t i k +1 ) along with the decentralized triggering condition 2   � e 2 i ≤ σ i γ i ( t ) − γ j ( t )  j ∈N i achieves consensus for the single integrator agents. Here e i ( t ) := [ L γ ( t i k )] i − [ L γ ] i and t i k is the event time for the agent i . 0 < σ i < 1 is the tuning parameter.

Event-based Cooperative Control ◮ Given the dynamics of path variable γ i v i γ i = v d + ˜ ˙ r + g i ◮ The results of event-triggered consensus (practical) hold in pres- ence of v d and g i . That is, � v i ( γ i ( t i k ) − γ j ( t i ˜ r ( t ) = − k )) j ∈N i and 2   e 2 � i ≤ σ i γ i ( t ) − γ j ( t )  j ∈N i achieves synchronization of path variables γ i . Continuous measurement (communication)!

Step 2: Event-based Communication ◮ For a generic agent i , define the communication packet C i ( t i t i k , γ i ( t i v i r ( t i k ) , g i ( t i � � k ) := k ) , ˜ k ) Consequently, agent i receives C j ( t j k j ( t ) ) from j ∈ N i . v j r ( t ) is held constant over the time interval t ∈ [ t j k , t j ◮ ˜ k +1 ) for all j ∈ N i . Hence, agent i estimates, γ j ( t ) = γ j ( t j k j ( t ) ) + ( t − t j r ( t j k j ( t ) )) + g j ( t j v j ˆ k j ( t ) )( v d + ˜ k j ( t ) )) ◮ Then event is generated on agent i using, 2   � e 2 i ( t ) ≤ σ i γ i ( t ) − ˆ γ j ( t )  j ∈N i Result: Event-based communication!

Event-based Cooperative Path Following Event-based Cooperative Path Following Network Event-based ETC Consensus ZOH Path Following Controller Robotic Vehicle Cascade of two ISS subsystems!

Experiment Results ◮ Cooperative Path Following in cir- cular paths using three AUVs ◮ Constant speed assignment of v d = 0 . 035 [rad/s]. ◮ Sampling frequency of 100 Hz. 10 AUV1 AUV2 5 AUV3 ◮ Gains of Path Following tuned man- Longitundinal error [m] 0 ually, ǫ = [0 . 3 0] T . -5 -10 -15 -20 15:20:00 15:25:00 15:30:00 Time [HH:MM:SS] 10 AUV1 8 AUV2 AUV3 6 Lateral error [m] 4 2 0 -2 -4 15:20:00 15:25:00 15:30:00 Time [HH:MM:SS]

25 Experiment Results 1.5 20 1 0.5 15 0 Path variable ◮ Error between the path variable γ i 15:20:00 10 { 1 , 2 , 3 } of each AUV for i = 5 AUV1 changes speed γ 1 asymptotically converges to zero. 0 γ 2 γ 3 -5 Consensus!! 15:20:00 15:25:00 15:30:00 Time [HH:MM:SS] 1.2 AUV1 ◮ ˙ γ i → v d . Desired speed assignment AUV2 1 AUV3 AUV1 starts achieved. 0.8 AUV2 starts AUV3 starts 0.6 γ i ˙ 0.4 0.2 X: 7.37e+05 Y: 0.03524 Table 1 : Event time for Circular formation 0 -0.2 AUV-1 AUV-2 AUV-3 15:20:00 15:25:00 15:30:00 Time [HH:MM:SS] Duration [s] 617.96 643.48 648.17 Max τ k [s] 160.24 32.10 78.80 15:26:00 Min τ k [s] 0.70 0.03 0.61 AUV1 15:25:00 AUV2 Num Events 31 36 51 AUV3 15:24:00 Periodic 61796 64348 64817 Time [HH:MM:SS] % Comms 0.050 0.055 0.078 15:23:00 15:22:00 15:21:00 More Events 15:20:00 15:19:00 15:18:00 15:20:00 15:25:00 15:30:00 Time [HH:MM:SS]

Open Problems You want to communicate, but cannot?? ◮ Preliminary tests show that the proposed event-based method can tolerate communication losses. ◮ Formal investigation needed to analyze effects of communica- tion/packet losses and communication delays. ◮ Delays can play important role in underwater acoustic commu- nications. Different Formation Control approaches?? ◮ The current approach → Static formations! ◮ Can the formations be more dynamic?

Questions???