After introducing the advanced navigation and control techniques that served as the motivation for this work, the challenges and scope of research related to the precedent targets have been identified. Traditionally, the guidance and control of pursuit vehicles has been achieved using first-order gradient descent along the surface of an artificial potential field.
Prelude
- Outline of Chapter 2
- Outline of Chapter 3
- Outline of Chapter 4
- Outline of Chapter 5
- Outline of Chapter 6
Efficient guidance and control techniques for driverless cars [30], evasive target pursuit [28], multi-agent tracking [31], sensor network-based decentralized tracking [35], and software robotic applications [29] have been reported. An integrated speed control controller [58] has been reported to take advantage of this knowledge to compute smooth cubic spline curves, while another integrated controller [59] is said to combine proportional speed control with differential braking.
Overview of Key Contributions
Khosla, “Real-time obstacle avoidance using harmonic potential functions,” IEEE Transactions on Robotics and Automation. Yang, “Optimal Pure Proportional Navigation for Target Maneuvering,” IEEE Transactions on Aerospace and Electronic Systems.
Optimal Problem Formulation
Linear Search Direction
If there is only an isolated global minima (the target) and no obstacles, the potential surface appears flat with a point trough and the level curves are represented by concentric circles.1 Figure 2.1(a) confirms that the magnitude of the potential decreases linearly to zero as the search direction develops along the negative orthogonal direction with respect to the level curves. However, with obstacles, the level curves become irregular in shape and the search direction is no longer a straight line.
Smoothness, Convergence and Solution
Problems with Line Search
Controlled by the irregularity in potential terrain, the pursuer's trajectory turns out to be oscillating, which can be seen in Figure 2.1(b).
Curvature of Potential Surface
The Levenberg modification may fail to preserve the curvature along the eigenvectors of the true Hessian. The eigenvectors corresponding to the eigenvalues belong to the asymptotes of the curvature of the potential surface.
Comparative Study
Case Study 1: Sparse Environment
The trajectory is found to be further smoothed after applying the Marquardt modification (Figure 2.5(a)) for a comparative latency efficiency (34 iterations required for convergence) (see Figure 2.5(b)). It is noted that the latter technique is able to handle a larger value of μ that preserves the curvature information of the potential surface.
Case Study 2: Populated Environment
The performance of the first and second order path search routines are summarized in Table 2.1. The Levenberg-Marquardt algorithm through the obstacle set in both conditions are illustrated in Figure 2.8(a) and Figure 2.8(b), respectively.
Summary of Chapter 2
In the present context, tracking ends with an intercept, which refers to a state where the target and follower trajectories occupy the same set of states and/or control variables at a single instant. On the other hand, if the tracker is scheduled to intercept the moving target at the end of a predetermined time interval after the start of tracking, it refers to a definite finite horizon problem.
Kinematic Assumptions
Free Finite-Horizon Problem
- Problem Formulation
- Necessary first-order conditions
- Determination of Boundary Values
- Determination of Free Terminal Time
- Determination of The Constants
Assuming that the speed of the pursuer is non-zero for all ∈[0, T] and rearranging the terms in equation (3.5), the control variables can be expressed explicitly in terms of co-states as shown in equation (3.6). Interception constrains the pursuer's terminal states to be a subset of the target trajectory at the terminal time, T.
Observations and Analysis
Role of Weighing Parameters
Now, referring to Table 3.2, we can see that increasing speeds up the process with T(s) Total Cost % Final detection error. The calculated value of time-to-interception increases with an increase in wv, as does the total cost shown in Table 3.3.
Performance Relative to Non-optimal Techniques
The line of sight method [13] is a reactive technique that iteratively minimizes the line of sight distance between the target and the pursuer. Statistics showing the final tracking error (see Table 3.4) achieved by the different methods discussed here confirm the effectiveness of our proposed optimal technique.
Tracking with Fixed Terminal Time
Selection of Design Parameters: Tracking
The process of determining the length of the fixed horizon, Tint, is heuristic, but not entirely arbitrary. Individual sizes of the weights, wd (weight to tracking error) and wv (weight to control effort) were observed to directly affect the cost incurred.
Summary of Chapter 3
Meng, “Neural network approaches to collision-free dynamic trajectory generation,” IEEE Transactions on Systems, Man, Cybernetics: B, Cybernetics. Ghose, “True Proportional Navigation with Maneuvering Objective,” IEEE Transactions on Aerospace and Electronic Systems.
Concept of Decoupled Controllers
Obstacle: Assumptions and Representation
The benefits provided by 'buffer space' of the representative obstacle are pronounced if the obstacle shows unexpected movement. Care was taken to minimize waste of reachable space while constructing the representative obstacle.
Problem Formulation
This step plays a critical role in eliminating or at least minimizing the possibilities of collision caused by improper timing of the second coupling event. In the latter case, it helps to minimize tracking error due to the optimal location of the pursuer at the time of the second shift.
Optimal Conditions and Trajectories
Smoothening Filter
The filter uses a 'mixing function', α(t) given in equation Here t = (ts−τ) indicates the start of the smoothing process and τ is the duration of the operation (ts is the switching instant). It depends on the difference in the levels of the control variable and the slew rate.
Observations and Analysis
Selecting the Design Parameters
The most important design parameter in the anti-collision mode of navigation is the duration of avoidance. Choice of an optimal avoidance duration, ∆ is made before t3, which marks the first switching moment from the tracking to the anti-collision mode.
Smooth Trajectory Design at Switching
Therefore, the reconstructed states within the smoothing interval do not make an exact fit within the asymptotic references provided by the original trajectories. The percentage of deviation from the desired optimal path represented by the sub-optimal smoothing filter is within the accepted dynamic limits.
Comparison of Performance
On the other hand, experiments show that NIC generates a very relaxed trajectory due to the trade-off between the tracking and collision avoidance subtasks. A large portion of the safely available set is unnecessarily omitted due to this trade-off, and the track is observed to move dangerously close to the wall.
![Figure 4.4: Comparison of trajectories from different guidance methods. SCBC [5] is a control barrier function based method and NIC [4] is an integrated op-timization and control method](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10342093.0/96.918.259.689.110.379/figure-comparison-trajectories-different-guidance-function-integrated-timization.webp)
Summary of Chapter 4
Huntsberger, "Safe Maritime Autonomous Navigation With COLREGS, Using Speed Barriers," IEEE Journal of Oceanic Engineering. Savkin, "Circular Navigation Missile Guidance with Incomplete Information and Uncertain Autopilot Model," IEEE Journal of Guidance.
Foreword
Integrated Controller Design
Optimality Conditions
Instead of the traditional quadratic form of control energy [3], this problem uses a logarithmic speed barrier. Stationarity' of the Hamiltonian with respect to variations in the steering vector leads to equation (5.7), which expresses the steering variables in terms of the .
Boundary Value Problem and its Solution(s)
Inaccuracies of the initial iteration can be overcome as new measurements become available over time. This means that the initial states for the augmented state vector are different for each solution.
Logarithmic Penalty Function
It imposes a penalty if the value function of the argument varies between (0,1), with penalty increasing exponentially as the value tends to zero. It can be used for attitude inference based speed control of the pursuer while dealing with obstacles of different speeds, numbers, dimensions and approach angles.
Results and Analysis
Selection of Weighing Constants
The weight constants wd, wb, and wv are design parameters, configured to assign relative priorities to tracking, collision avoidance/steering, and speed regulation/control effort, respectively. A higher wv reduces both control effort and peak velocity (PV) achieved, but relaxes the minimum safety margin (MSM) from the obstacle(s).
Logarithmic Barrier : Steering and Velocity Control
The collisions reported for configuration 7 (bold values are collisions) are evident because the control weights are both low and equal. c) The following example demonstrates a simulated tracking scenario that implements the shrinking horizon model predictive control policy. A single static obstacle at (−0.5.2)m is avoided by adjusting the steering weight to 0.3 and maintaining a moderate velocity barrier of 1m/s.
Comparison With Other Methods
The tracking error (see Figure 5.6(b)) and the velocity profiles of the pursuer, target and obstacles can be seen in Figure 5.6(c) for a better understanding of the above description. It will further be seen from the acceleration (Figure 5.9(c)) and angular velocity (Figure 5.9(d)) profiles of the pursuer illustrated in Figure 5.9 that our proposed method is more effective in providing passenger comfort due to reduced fluctuations and the size of the control variables.
Summary of Chapter 5
Zhao, “Optimal path planning and speed control integration strategy for UGVs in static and dynamic environments,” IEEE Trans. Real-time motion planning for aerial imaging with dynamic obstacle avoidance and viewpoint optimization,” IEEE Robotics And Automation Letters.
Closed-loop Tracking Controller
Stability of Feedback Tracking Control
This example shows the evolution of state and control errors over a 3 s interval for a trajectory segment. A higher value of ˆm speeds up convergence, but this comes at the expense of a higher initial size of the local control variables.
Wheel-Slip as Disturbance
Observer Design
Linear Observer
The observer error, e is a difference vector of the actual and the estimated conditions and is represented by q−q. For the same reason, the observer error can be designed to converge asymptotically by selective pole placement of the matrix (A −LC), regardless of the controller design, (A−BK) (K is the controller gain).
Nonlinear Observer
However, the drift of the inertial measurement results in incremental errors over time and must be updated regularly. In the absence of an oscillation velocity meter (magnetometer), the direction provided by the optical current was assumed as the update reference.
Slip Propagation Model and Modified Kinematics
A discrete-time linearization of (6.15) can now correctly predict the attainable state in the next instant. It can be recalled again that the proposed method aims to facilitate an accurate estimation of condition in the presence of sliding disturbance and has no direct relation to the controller used or the quality of measurement.
Experiments and Simulations
- Case Study : Changes in Linear Velocity
- Case Study : Turning manoeuvres
- Effectiveness of Slip-kinematic Model
- Selection of Observer Gain
The slip angle reaches a maximum in the mentioned interval (see Figure 6.6(b)) and changes sign due to the transition from understeer to oversteer. The close agreement between the observed slip parameters (slip angle in Figure 6.9(a) and slip ratio in Figure 6.9(b)) and the estimated slip parameters proves the efficiency of the estimation process.
Summary of Chapter 6
Darouach, “Convergence analysis of an extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems,” IEEE Transactions on Automatic Control. Barrau., “Invariant Extended Kalman Filter as a Stable Observer,” IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers.
Obstacle Mapping
The point corresponding to the centroid of the resulting planar polygon forms the center of the representative jth obstacle, (xjobs(t), yobsj (t)). Then assume that the line segment connecting the center of the ith obstacle and its farthest vertex is called ri.
Pursuer Model
Reduces computational footprint: This minimal state-space representation helps in faster runtime execution of iterations of the optimization routine in question. Finite speed limit: This model inherently imposes a finite limit on the forward linear speed, validating the applicability of the maximum principle.
State Estimation : Fixed Finite Horizon
The function, f(y,y, t) is continuous over the domain of˙ f(y,y, t), at least˙ within the interior of the domain. Lipschise property can be verified in a number of ways depending on the nature of the function f(y,y, t), some require evaluation of one or more pairs of constants, while some require the verification of boundedness of partial derivatives. , Dy, Dy˙, Dt of the function f(y,y, t).
Existence and Uniqueness: Tracking
We can choose any pair of non-negative Lipschitzian constants, Kx and Ky such that |wo|≤Kx and |wo|≤Ky [3]. If it is possible to choose a positive real intercept time such that, Tint As a result, we get a pair of second-order ordinary differential equations (8.7) in terms of the variables ˜x(t) and ˜y(t), where p=wλ3. Therefore, f(t,x) and˜f(t,y) are Lipschitzian since it is always possible to choose two˜ non-negative constants, Kx and Ky, such that |p|≤Kx and |p|≤ Ky. It is clear that the size of the interval, [0, Hue] plays a crucial role in satisfying the unique evolution of states. Based on these assumptions, our task is limited to verification of the Lipschitz condition by checking the boundary property of the partial derivatives of the expressions in the right-hand side of (5.8). Each non-zero element of ∆f1 can be expressed as a rational fraction, whose denominator polynomial has a greater degree than that of the numerator, which implies, the partial derivative with respect to tos(t) has a finite limiting value, and that the function is bounded. Waltman, "Existence and uniqueness of solutions of boundary value problems for two-dimensional systems of nonlinear differential equations," Transactions of the American Mathematical Society. The quadrature encoders measure the pose and the inertial measurement unit measures the vehicle's acceleration and angular velocity. For target, obstacle and follower localization, various pre-registered visual markers are attached to the tip, each with a unique and asymmetric pattern. Length of update horizon is determined by the data rate of the slowest sensor (for example camera). In case of a malfunction of the client-server communication, the current plan is terminated and the horizon update reinitializes the trajectory planner. The state feedback is used to calculate the tracking error and stabilizes the trajectory planner's open-loop data flow. The transformation matrix is designed according to the height, tilt and pan of the camera mount. Kim, “A UDP-based protocol for mobile robot control over wireless Internet,” Proceedings of ROBOCOMM'09: IEEE International Conference Robot Communication and Coordination. Zhang, “An embedded wireless transmission system based on the extended user datagram protocol (EUDP),” Proceedings of ICFCC'10: IEEE International Conference on Future Computer and Communication.Existence-Uniqueness: Plan-B
Existence: Integrated Approach
Derivation of Slip-ratio Propagation Model
Hardware Description
Control Architecture
Software Tools
Algorithms
