2.4 Comparative Study
2.4.2 Case Study 2: Populated Environment
The next case study pertains to a densely populated environment in contrast to the sparsely populated flat potential surface featured in the preceding discussion.
The potential field is characterized by a target close to a narrow valley surrounded by a cluster of stationary obstacles of radius 1m each, positioned at (−2,0) m,
−10 −8 −6 −4 −2 0 2 4
−10
−5 0 5
Distance along x-axis [m]
Distance along y-axis [m]
0 5 10 15 20 25 30 35
0 2 4 6 8 10 12
Number of steps
Distance [m]
Figure 2.5: Performance evaluation of Levenberg-Marquardt Method in flat po- tential surface showing (a) smooth trajectory and (b) rate of convergence of dis- tance function between the target and the pursuer.
(−4,2) m, (−5,0) m and (−3,2) m. The target is located at (−5,−2) m and as- sumed to retain same position during tracking. The idea behind the static setting is to study the tracking performance of the first and the second order gradient based path planners when the pursuer is compelled to traverse through the narrow valley and simultaneously compute subsequent search directions. This means, we purposefully design the problem such that the pursuer does not have any option to bypass the obstacle cluster. The pursuer is assumed to start tracking from the position, (3,7) m.
Linear gradient descent achieves interception in 764 iterations as shown in Fig- ure 2.6(a). Figure 2.6(b) illustrates the trajectory of the robot for the same navigation planner. Adding momentum term helps to achieve convergence in 65% less time, within 271 iterations. Levenberg method with µ=0.3 achieves convergence in 211 iterations, while Levenberg-Marquardt algorithm yields the same result in only 160 steps. In the following simulations an upper limit of 1000 iterations have been considered as admissible threshold for successful intercep- tion. Interception is assumed to be achieved if the distance function between the target and the pursuer is less or equal to 0.1m. Although analysis of results of the previous simulation confirms the faster convergence property of quadratic search routines over linear search, true efficiency of the second order optimization meth-
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Obstacles Start Target
Distance along x-axis [m]
Distance along y-axis [m]
0 200 400 600 800
0 5 10 15
Number of steps
Distance [m]
Figure 2.6: Performance evaluation of linear gradient descent in narrow valley (a) trajectory of pursuer (b) rate of convergence of distance function between the target and the pursuer.
ods may be observed in scenarios involving narrower passages. Passage through the obstacle cluster is further narrowed by 23% to a spacing equal to the radius of influence of an obstacle. This means, all four obstacles need to be compulsorily avoided. Under this stringent path constraint, Levenberg-Marquardt algorithm
0 100 200 300 400 500 600 700 800 900 1000 2
4 6 8 10 12 14
Number of steps
Distance [m]
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−10
−5 0 5
Target
Obstacles Start
Distance along x-axis [m]
Distance along y-axis [m]
Figure 2.7: Finding optimal solutions with first and second order search direc- tions in narrow valleys (a) trajectory computed with Newton’s direction fails to converge within admissible threshold of iterations (b) trajectory fails to converge with line search.
(LMA) is found to successfully complete the task in 187 iterations, for a weigh- ing factor chosen as 0.3. Modified Newton’s mehtod (NM) or Levenberg method fails to reach the target within the predefined admissible limit of steps as shown in (Figure 2.7(a)). On the other hand, (Figure 2.7(b)) illustrates the pursuer’s trajectory under linear gradient descent. The task remains incomplete as the tracking trajctory fails to satisfy the terminal distance function criterion for vali-
dating inteception. In both of these cases, it is concluded that no solution exists through a very narrow passage for all planners except the Levenberg-Marquardt technique. The performances of the first and the second order path search rou- tines have been summarized in Table 2.1. Succesful execution of tracking by the
Method Flat Gradient Narrow Valley
GD 98 764
GDM 42 271
NM 38 211
LMA 34 160
Table 2.1: Comparative study of number of iterations required for convergence by first and secord order search directions in a narrow populated environment.
GD: Gradient Descent, GDM: Gradient Descent with momentum, NM: Newton’s Method or Levenberg’s Method, LMA: Levenberg-Marquardt Algorithm
Levenberg-Marquardt algorithm through the obstacle cluster in both conditions have been illustrated in Figure 2.8(a) and Figure 2.8(b) respectively. However, as
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Distance along x-axis [m]
Obstacles Target
Start
Distance along x-axis [m]
−12 −10 −8 −6 −4 −2 0 2 4
−10
−5 0 5
Distance along x-axis [m]
Start
Obstacles Target
Distance along y-axis [m]
Figure 2.8: Performance evaluation of Levenberg-Marquardt in populated envi- ronments (a) trajectory in a narrow valley (b) trajectory when reachable set is further reduced.
pointed out in [4, 7], the second order methods are sub-optimal. Linear gradient descent is globally convergent but slow and inefficient in both sparsely populated and densely populated situations. Whereas, convergence of the quadratic meth-
ods can be guaranteed only if the tracking is assumed to start within a small neighbourhood of the target position (minima). For any arbitrary initial point, this refers to a continuous evaluation of search directions upon applying a change of variables. Intensive computations pertaining to calculation of true Hessian and its inverse in quadratic optimization routines may be bypassed by said approx- imation. According to Ren et al. [2] this approximation is feasible for systems with two dimensional state space. This further encouraged us to define the kine- matics of the pursuer by a two-state, two-input model, to be explained in details in the next chapter.
Gradient based planners are global optimization algorithms that may suffer from possible loss in continuity of trajectory because of local minima traps generated by dynamic distribution of obstacles and target. Recently, a noise-free, reactive nav- igation technique has been proposed in [9] wherein, the pursuer is driven towards an extremum without computation of state derivatives. But the results indicate useful outcome only for well-behaved targets and the article remains oblivious to movement of obstacles in the environment. Another controller proposed in [10]
uses Harmonic potential vector-field that matches a navigation function to the kinetic energy of the vehicle, but does not take into account issues caused by mobility of obstacles. Besides, pursuit vehicles such as guided carts, industrial inspection and surveilance robots may benefit from multiple other parameter opti- mizations for cost-effective performance and safety requirements, which cannot be achieved by gradient based guidance alone. Hence customizable optimal planners have been developed in Chapter 3. It will be shown in the following chapters that multi-objective optimal controllers are better equipped to planning with relative weightage and generate smoother trajectories for a variety of dynamic situations.