# Projects

Consensus is a scenario where multiple agents interact with each other to converge asymptotically towards a common value. In recent times, convergence analysis of consensus protocols has drawn significant research attention among control researchers. However a large part of convergence analysis is based on a time-invariant graph structure. Our work [1], provides bounds on the convergence rates to consensus that hold for a large class of time-varying edge weights. A novel analysis approach based on classical notions of persistence of excitation and uniform complete observability has been developed. Furthermore in [2], a novel application of results from Ramsey theory allows for proof of consensus and convergence rate estimation under a switching graph topology. Sample spanning tree graphs and the corresponding convergence of spanning tree edges to consensus along with a rate envelope are shown below.

Event-triggered control is a new paradigm where discretization of control, instead of being periodic is allowed to be state-dependent and activated via a trigger function. This is seen as an energy efficient means of control design. In [10], we investigate the affects of event-triggered control on consensus under switching graph networks and double-integrator agent models. This extends earlier work that assumes static network models. ifac_poster.pdf

Orientation (Attitude) and position control of spacecraft are a key component of mission objectives for all space missions. Attitude control is primarily used to ensure that pointing requirements of sensors, solar arrays and other science equipment are satisfied. Synchronisation of multiple spacecraft in orbit is vital for interferometry applications among others. Our group's work proposes several feedback laws to ensure precise position formation and attitude synchronisation for cooperative applications. The novelty of our results with relation to literature is two-fold, i) we design coordinate free control making use of the rotation matrix directly in design, ii) the control laws require only line of sight measurements which are easily available and not dependent on absolute attitude measurement.

Line-of-sight based control laws for attitude alignment along line-of-sight and formation keeping of two spacecraft in formation are proposed in [3]. The control objective is to achieve, attitude alignment about the LOS vector between the two spacecraft and a desired relative distance between the spacecraft. Proposed control laws are distributed in nature and are obtained in respective body frames. Desired equilibrium configuration under closed loop dynamics are shown to be locally asymptotically stable with a conservative region of attraction specified. The work is extended in [2] to complete attitude synchronization and formation keeping of two spacecraft with a leader using line-of-sight measurements. Each spacecraft measures line-of-sight unit vectors to each other and to the leader spacecraft in respective body frames. These line-of-sight measurements are communicated to each other. Lyapunov-like stability analysis is used to prove stability of the desired equilibrium configuration under closed loop dynamics. The work is further extended to relative attitude trajectory tracking using line-of-sight measurements in [5], [6], and [7], where the objective is to control attitudes of the spacecraft so that their relative attitude tracks a desired time-varying relative attitude trajectory. In [5] a distributed architecture is followed, while in [6] and [7] serial communication architecture are followed. The attitude control laws are obtained in terms of line-of-sight vectors and guarantee asymptotic tracking of desired attitude trajectories even in the presence of spacecraft translation dynamics under gravity. No observations to external objects such as stars are used, which typically require expensive sensor equipment and complex on-board computation algorithms.
The rotation matrix is a global representation of attitude and therefore precludes the possibility of winding like phenomena typical with parametrizations affecting uniqueness of solutions. Further, line of sight based control laws minimize the need for expensive star sensors and associated computation for absolute attitude estimation. The results in [3]-[7], therefore have great utility in micro-satellite based multiple spacecraft missions.

Coverage control refers to design of feedback laws for multiple robotic agents to optimally cover a convex region of interest. The region is typically of interest in this context due to unfolding of a phenomena that requires distributed sensing, such as an oil spill, nuclear radiation etc. Optimal coverage control involves distributing agents in the region based on the intensity of the sensory function. The centroidal Voronoi configuration has been shown in literature to be optimal in such applications. Our research extends literature in several ways, i) we consider dynamics level model instead of kinematics as in literature for more accurate depiction of a real system, ii) we adapt for unknowns in the dynamics, such as inertia in addition to the sensory function as in literature to allow for uncertainties [8]. The following plots show sample initial and final positions using our coverage strategy with large peaks indicated by solid red dot and small peaks by hollow red dots.

In the above experimental investigations, the coverage control algorithms designed by us are applied to mobile ground robots. The phenomenon is the light generated by two LED's, one brighter and another dimmer. Two different algorithms were arrived at, the top video shows results based on the locational optimisation framework and the bottom video shows the results based on an $L_2$ minimisation of the distance between sensor and light densities. In both cases we notice that more robots settle closer to the higher intensity light as expected.

Energy efficiency has emerged as one of the key challenges in recent times for all applications. Event-triggered control is also seen as a means to minimize actuation in control systems. $L_0$ optimal control offers a natural way to design sparse controllers for dynamical systems. In contrast to $L_1$ optimal control laws in literature, the control magnitudes are precisely zero and not `close' to zero. The minimization is on the activation time. We have studied both the reachability and linear quadratic problems in this framework to obtain sparse controls for the same [9]. A comparison between the performance of the LQ optimal control and the discontinuous $L_0$ optimal control is shown below.

Gazebo simulation of the Backstepping law on an AR Drone model

Initial Hardware trial of the Backstepping law on the AR Drone

AR Drone following an RC car using a Back-stepping algorithm

[1] N. Roy Chowdhury and S. Sukumar, “Persistence based analysis of consensus protocols for dynamic graph networks,” in the proceedings of European Control Conference (ECC), 2014, pp. 886-891.
[2] N. R. Chowdhury, S. Sukumar, and N. Balachandran, “Persistence based convergence rate analysis of consensus protocols for dynamic graph networks.” European Journal of Control, 2016.
[3] Warier, R. and Sinha, A., “LOS based attitude alignment of two spacecraft in formation,” 5th International Conference of Spacecraft Formation Flying Missions and Technologies, Munich, 2013.
[4] Warier, R., Sinha, A. and Srikant, S., “Spacecraft Attitude Synchronization And Formation Keeping Using Line Of Sight Measurements,” Proceedings of 19th IFAC World Congress, Cape Town, 2014.
[5] Warier, R., Sinha, A. and Srikant, S., “Relative Attitude Trajectory Tracking Using Line of Sight Measurements under Spacecraft Position Dynamics,” Advances in Control and Optimization of Dynamical Systems, Vol.3, No. 1, 2014, pp.455-461. doi: 10.3182/20140313-3-IN-3024.00236
[6] Warier, R., Sinha, A. and Srikant, S., “Line Of Sight Based Spacecraft Formation Control Under Gravity,” Indian Control Conference, Chennai, 2015.
[7] Warier, R., Sinha, A. and Srikant, S., “Line-of-sight Based Attitude and Position Tracking Control For Spacecraft Formation,” European Journal of Control, 2016. doi: 10.1016/j.ejcon.2016.04.001
[8] Razak R. A. , Srikant S. and Chung H., “Decentralized Adaptive Coverage Control of Nonholonomic Mobile Robots” , 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), August 23-25, 2016
[9] Srikant S. and Chatterjee D., “A jammer's perspective of reachability and LQ optimal control”, Automatica , Vol. 70, August 2016, Pages 295–302
[10] S. Arun Kumar, N. R. Chowdhury, S. Srikant, J. Raisch, “Consensus Analysis of Systems with Time-varying interactions: An Event-triggered Approach”, 20th IFAC World Congress, Toulouse, France, July 2017