Understanding Complex Systems May 15, 2007 Javier Alcazar, Ph.D. Minimum Time Control and Time Estimation to Intercept Mobile Targets Under Robot Dynamic Constraints Understanding Complex Systems May 15, 2007 Javier Alcazar, Ph.D. Javier Alcazar

Problem Description Given 1 linearly moving, constant speed target 1 robotic pursuer 2D surface with no bounds and no obstacles What are the possible set of pursuer maneuvers, to intercept the mobile target in the minimum time? What is the estimated amount of time to intercept target? What controller allows the minimum interception time? Javier Alcazar

Related Previous Work Balkcom, D. J., and Manson, M. T. “Time optimal trajectories for differential drive vehicles,” International Journal of Robotics Research, vol. 21, no. 3, pp. 199-217, 2002 Presented a classification of time optimal trajectories for a 2 wheeled nonholonomic robot. Kinematics are used to construct the set of optimal maneuvers for a particular pair of start and finish static points. Kalmar-Nagy, T., Ganguly, P. and D’andrea, R. “Near-optimal dynamic trajectory generation and control of an omnidirectional vehicle”, Robotics and Autonomous Systems, vol. 46, pp. 47-64, 2004 A 3 wheeled omni directional robot is analyzed. Coupled control is assumed to decoupled, and the near-optimal control is designed for a particular pair of static start and finish points. No set for possible maneuvers. No target estimation. Stephen-Yeung, M. K. and Strogatz, S. H. “Nonlinear dynamics of a solid-state laser with injection”, Journal of Physical Review E, vol. 58, p.4421, 2000 Introduces dimensionless representation for the dynamics of a solid-state laser. Javier Alcazar

Related Previous Work Williams, R. L., Carter, B. E., Gallina, P. and Rosati, G. “Dynamic model with slip for wheeled omnidirectional robots.” IEEE Transactions on Robotics and Automation, vol. 18, no. 3, 2002 Use the redundancy of having 3 omnidirectional wheels on the robot, to give a prediction of the wheel that slips. Cliff, D. and Miller, G. F. “Co-evolution of pursuit and evasion II: Simulation Methods and results”, Simulations on Adaptive Behavior, 1996 Used a Genetic Algorithm to evolve the brain (Neural Network Controller) of the pursuer. Off-line method to capture prey focus on evolving where the observance sensors are placed. It has particle dynamics. No set of possible maneuvers or time estimators. Balkcom, D. J., Kavathekar, P. A. and Manson, M. T. “The minimum time trajectories for an omni-directional vehicle.” Algorithmic Foundations of Robotics, 2006 Presented a classification of optimal trajectories for a 3 wheeled omni directional robot. The problem of determining which of these trajectories is optimal for a particular pair of start and finish configurations is not addressed. Maximum control effort is applied, trajectories are obtained. Javier Alcazar

Related Previous Work Ghose, K., Horiuchi, T. K., Krishnaprasad, P. S. and Moss, C. F. “Echolocating Bats Use a Nearly Time Optimal Strategy to Intercept Prey”, In Press PLoS Biology 2006 Used big brown bats (Eptesicus fuscus) to study sonar guided flight. A complete insect chase from detection (about 3m from insect) to capture typically takes less than one second. Bat chased a single prey in the room (7.3m x 6.4m x 2.0m). No dynamics for the pursuer are presented. Experimental work mainly. No time estimators, no set for possible maneuvers. Javier Alcazar

Outline Kinematics and dynamics for the robotic pursuer Minimum time controller design Time estimators to intercept target Possible set of pursuer maneuvers Simulations and Results Conclusions Javier Alcazar

Kinematics and dynamics - The dynamics of each DC motor can be described by, - Since the electrical time constant of the motor is very small comparing to the mechanical time constant, dynamics of the electric motor can be neglected, i.e. and . - The torque produced by the DC motor can be described by, - Assuming that the wheels do not slip, the force generated by a wheel attached to the DC motor is given by, Javier Alcazar

Kinematics and dynamics - Positions where each wheel makes contact with the ground are calculated as, - Velocities where each wheel makes contact with the ground are calculated as, - Individual wheel velocities are given by the projection of along the unitary vector rotated by as follows Javier Alcazar

Kinematics and dynamics - Translational velocities of the wheels, - Using Newton’s law for the linear and angular momentum of the robot yields, Non linear, input coupled, differential equations. Javier Alcazar

Kinematics and dynamics - Dimensionless representation, define - Position and angular orientation of the robot is given by, Non linear, input coupled, differential equations, subject to Javier Alcazar

Minimum time control: Feedback Linearization - Transform the non-linear robotic model into a linear system using nonlinear feedback. - Position and angular orientation of the robot is given by, Linear in (x, y), (x, y) input decoupled, differential equations, subject to Javier Alcazar

Minimum time control: Strategic final velocity - System fully decouples into two independent identical equations - The analysis of the x coordinate also solves the expression for y. - A translation of coordinates for mathematical convenience is introduced as, Javier Alcazar

Minimum time control: Strategic final velocity - Translation of coordinates - Problem becomes: Find the time optimal control for transferring the state space system from an arbitrary initial state to the new origin Javier Alcazar

Minimum time control: Strategic final velocity THEOREMS Kirk [14].: Given the standard linear system S as, - THEOREM 1. (Existence) If all the n-eigenvalues of A have nonpositive real parts, then an optimal control exists that transfers any initial state to the origin. - THEOREM 2. (Uniqueness) If an optimal control exists, then it is unique. - THEOREM 3. (Number of switchings) If the n-eigenvalues of A are all real, and a (unique) time optimal control exists, then each control component can switch at most (n-1) times. Javier Alcazar

Minimum time control: Strategic final velocity - System fully decouples into two independent identical equations - Solving the differential equations with q1 constant yields, Javier Alcazar

Minimum time control: Strategic final velocity - The set , from which the new origin can be reached by applying , : - The set of points that are in either set is given by - The switching function is then given by, Javier Alcazar

Minimum time control: Strategic final velocity The minimum time control law is: i.e. for points “above” the switching curve the optimal control law is until reaching the switching curve, where switches to +1, and remains at +1 until the new origin is reached, at which time is applied to keep the system at the new origin. Javier Alcazar

Minimum time control: Without strategic final velocity The minimum time strategy for pursuing the mobile target, is to apply the maximum voltage possible towards the estimated future position of the target Switching functions are given by, Javier Alcazar

Time estimation to intercept target: Without strategic final velocity - Target is described by, - Target is assumed to be moving with constant x and y velocities, which implies that no forces are being applied, - The analysis of the x coordinate also solves the expression for y. - Recast into dimensionless form. - Initial conditions for the target - The position and velocity of the target are Javier Alcazar

Time estimation to intercept target: Without strategic final velocity - Using two observations, we can estimate target velocity as, - The estimated target position as, - Dynamics of the robotic pursuer - Define as the time to intercept the target on its x coordinate, interception will occur when i.e., Javier Alcazar

Time estimation to intercept target: Without strategic final velocity Solving for yields the “Algorithm to Compute the estimated time to intercept the target” If THEN ELSE { If THEN If THEN ELSE } Javier Alcazar

Time estimation to intercept target: With strategic final velocity Compute and then substitute in, where Javier Alcazar

Maneuvers With strategic final velocity, the pursuer performs no switching, the pursuer performs one switching in x, the pursuer performs first a switch in x, then a switch in y. Javier Alcazar

Maneuvers With strategic final velocity, the pursuer performs no switching, the pursuer performs one switching in x, the pursuer performs first a switch in x, then a switch in y. Without strategic final velocity, - Same sketch - The maneuvers acknowledge the inertia of the robot vehicle Javier Alcazar

Simulations and results Computer simulation showing, the target trajectory, in gray, and several pursuer trajectories, in black. The initial target position is , and several initial pursuers positions given by, (a) , (b) , (c) , (d) , and (e) . The initial pursuer velocity in each simulation was set to be , and initial target velocity Javier Alcazar

Conclusions Design equations were presented that could assist in the design of minimum time bang-bang controllers. Algorithms to compute the estimated time to intercept a mobile target with constant acceleration were presented. The feedback linearization proposed greatly simplified the design of the controller. The strategy aims towards the estimated position at the estimated time for interception. Javier Alcazar

Thank you Javier Alcazar