{"title": "Oscillatory Neural Fields for Globally Optimal Path Planning", "book": "Advances in Neural Information Processing Systems", "page_first": 539, "page_last": 546, "abstract": null, "full_text": "Oscillatory Neural Fields for \n\nGlobally Optimal Path Planning \n\nMichael Lemmon \n\nDept. of Electrical Engineering \n\nUniversity of Notre Dame \nNotre Dame, Indiana 46556 \n\nAbstract \n\nA neural network solution is proposed for solving path planning problems \nfaced by mobile robots. The proposed network is a two-dimensional sheet \nof neurons forming a distributed representation of the robot's workspace. \nLateral interconnections between neurons are \"cooperative\", so that the \nnetwork exhibits oscillatory behaviour. These oscillations are used to gen(cid:173)\nerate solutions of Bellman's dynamic programming equation in the context \nof path planning. Simulation experiments imply that these networks locate \nglobal optimal paths even in the presence of substantial levels of circuit \nnOlse. \n\n1 Dynamic Programming and Path Planning \n\nConsider a 2-DOF robot moving about in a 2-dimensional world. A robot's location \nis denoted by the real vector, p. The collection of all locations forms a set called the \nworkspace. An admissible point in the workspace is any location which the robot \nmay occupy. The set of all admissible points is called the free workspace. The \nfree workspace's complement represents a collection of obstacles. The robot moves \nthrough the workspace along a path which is denoted by the parameterized curve, \np(t). An admissible path is one which lies wholly in the robot's free workspace. \nAssume that there is an initial robot position, Po, and a desired final position, p J. \nThe robot path planning problem is to find an admissible path with Po and p J as \nendpoints such that some \"optimality\" criterion is satisfied. \n\nThe path planning problem may be stated more precisely from an optimal control \n539 \n\n\f540 \n\nLemmon \n\ntheorist's viewpoint. Treat the robot as a dynamic system which is characterized \nby a state vector, p, and a control vector, u. For the highest levels in a control \nhierarchy, it can be assumed that the robot's dynamics are modeled by the dif(cid:173)\nferential equation, p = u. This equation says that the state velocity equals the \napplied control. To define what is meant by \"optimal\", a performance functional is \nintroduced. \n\n(1) \nwhere IIxil is the norm of vector x and where the functional c(p) is unity if plies \nin the free workspace and is infinite otherwise. This weighting functional is used \nto insure that the control does not take the robot into obstacles. Equation 1 's \noptimality criterion minimizes the robot's control effort while penalizing controls \nwhich do not satisfy the terminal constraints. \n\nWith the preceding definitions, the optimal path planning problem states that for \nsome final time, t\" find the control u(t) which minimizes the performance functional \nJ(u). One very powerful method for tackling this minimization problem is to use \ndynamic programming (Bryson, 1975). According to dynamic programming, the \noptimal control, Uopt, is obtained from the gradient of an \"optimal return function\" , \nJO (p ). In other words, Uopt = \\1 JO. The optimal return functional satisfies the \nHamilton-Jacobi-Bellman (HJB) equation. For the dynamic optimization problem \ngiven above, the HJB equation is easily shown to be \n\n(2) \n\nThis is a first order nonlinear partial differential equation (PDE) with terminal \n(boundary) condition, JO(t,) = IIp(t,) - p, 112. Once equation 2 has been solved \nfor the J O, then the optimal \"path\" is determined by following the gradient of JO. \nSolutions to equation 2 must generally be obtained numerically. One solution ap(cid:173)\nproach numerically integrates a full discretization of equation 2 backwards in time \nusing the terminal condition, JO(t,), as the starting point. The proposed numerical \nsolution is attempting to find characteristic trajectories of the nonlinear first-order \nPDE. The PDE nonlinearities, however, only insure that these characteristics exist \nlocally (i.e., in an open neighborhood about the terminal condition). The resulting \nnumerical solutions are, therefore, only valid in a \"local\" sense. This is reflected in \nthe fact that truncation errors introduced by the discretization process will even(cid:173)\ntually result in numerical solutions violating the underlying principle of optimality \nembodied by the HJB equation. \n\nIn solving path planning problems, local solutions based on the numerical integra(cid:173)\ntion of equation 2 are not acceptable due to the \"local\" nature of the resulting \nsolutions. Global solutions are required and these may be obtained by solving an \nassociated variational problem (Benton, 1977). Assume that the optimal return \nfunction at time t, is known on a closed set B. The variational solution for equa(cid:173)\ntion 2 states that the optimal return at time t < t, at a point p in the neighborhood \nof the boundary set B will be given by \n\nJO(p, t) = min {JO(y, t,) + lip - Y1l2} \n\n(t, - t) \n\nyeB \n\n(3) \n\n\fOscillatory Neural Fields for Globally Optimal Path Planning \n\n541 \n\nwhere Ilpll denotes the L2 norm of vector p. Equation 3 is easily generalized to \nother vector norms and only applies in regions where c(p) = 1 (i.e. the robot's free \nworkspace). For obstacles, ]O(p, i) = ]O(p, if) for all i < if. In other words, the \noptimal return is unchanged in obstacles. \n\n2 Oscillatory Neural Fields \n\nThe proposed neural network consists of M N neurons arranged as a 2-d sheet \ncalled a \"neural field\". The neurons are put in a one-to-one correspondence with \nthe ordered pairs, (i, j) where i = 1, ... , Nand j = 1, ... , M. The ordered pair \n(i, j) will sometimes be called the (i, j)th neuron's \"label\". Associated with the \n(i, j) th neuron is a set of neuron labels denoted by N i ,i' The neurons' whose labels \nlie in Ni,i are called the \"neighbors\" of the (i, j)th neuron. \nThe (i, j)th neuron is characterized by two states. The short term activity (STA) \nstate, Xi,;, is a scalar representing the neuron's activity in response to the currently \napplied stimulus. The long term activity (LTA) state, Wi,j, is a scalar representing \nthe neuron's \"average\" activity in response to recently applied stimuli. Each neuron \nproduces an output, I(Xi,;), which is a unit step function of the STA state. (Le., \nI(x) = 1 if X > 0 and I(x) = 0 if x ~ 0). A neuron will be called \"active\" or \n\"inactive\" if its output is unity or zero, respectively. \nEach neuron is also characterized by a set of constants. These constants are either \nexternally applied inputs or internal parameters. They are the disturbance Yi,j, \nthe rate constant Ai ,;, and the position vector Pi,j' The position vector is a 2-d \nvector mapping the neuron onto the robot's workspace. The rate constant models \nthe STA state's underlying dynamic time constant. The rate constant is used to \nencode whether or not a neuron maps onto an obstacle in the robot's workspace. \nThe external disturbance is used to initiate the network's search for the optimal \npath. \n\nThe evolution of the STA and LTA states is controlled by the state equations. These \nequations are assumed to change in a synchronous fashion. The STA state equation \nIS \n\nxtj = G (x~j + Ai,jYi,j + Ai,j L Dkl/(Xkl/\u00bb) \n\n(k,')ENi,i \n\n(4) \n\nw here the summation is over all neurons contained within the neighborhood, N i ,j , \nof the (i,j)th neuron. The function G(x) is zero if x < 0 and is x if x ~ O. \nThis function is used to prevent the neuron's activity level from falling below zero. \nDk/ are network parameters controlling the strength of lateral interactions between \nneurons. The LTA state equation is \n\nwT. = w:-\u00b7 + 1/'(xi J')I \n\nI,J \n\nI,} \n\nI \n\n(5) \n\nEquation 5 means that the LTA state is incremented by one every time the (i, j)th \nneuron's output changes. \n\nSpecific choices for the interconnection weights result in oscillatory behaviour. The \nspecific network under consideration is a cooperative field where Dkl = 1 if (k, I) i= \n\n\f542 \n\nLemmon \n\n(i,j) and Dkl = -A < \u00b0 if (k, I) = (i,j). Without loss of generality it will also be \n\nassumed that the external disturbances are bounded between zero and one. It is also \nassumed that the rate constants, Ai,j are either zero or unity. In the path planning \napplication, rate constants will be used to encode whether or not a given neuron \nrepresents an obstacle or a point in the free-workspace. Consequently, any neuron \n\nwhere Ai,i = \u00b0 will be called an \"obstacle\" neuron and any neuron where Ai,i = 1 \n\nwill be called a \"free-space\" neuron. Under these assumptions, it has been shown \n(Lemmon, 1991a) that once a free-space neuron turns active it will be oscillating \nwith a period of 2 provided it has at least one free-space neuron as a neighbor. \n\n3 Path Planning and Neural Fields \n\nThe oscillatory neural field introduced above can be used to generate solutions of \nthe Bellman iteration (Eq. 3) with respect to the supremum norm. Assume that all \nneuron STA and LTA states are zero at time 0. Assume that the position vectors \nform a regular grid of points, Pi,i = (i~, j~)t where ~ is a constant controlling the \ngrid's size. Assume that all external disturbances but one are zero. In other words, \nfor a specific neuron with label (i,j), Yk,l = 1 if (k, 1) = (i,j) and is zero otherwise. \nAlso assume a neighborhood structure where Ni,j consist of the (i, j)th neuron and \nits eight nearest neighbors, Ni,i = {(i+k,j+/);k = -1,0,1;1= -1,0,1}. With \nthese assumptions it has been shown (Lemmon, 1991a) that the LTA state for the \n(i, j)th neuron at time n will be given by G( n - Pk,) where Pkl is the length of the \nshortest path from Pk,l and Pi,i with respect to the supremum norm. \nThis fact can be seen quite clearly by examining the LTA state's dynamics in a \nsmall closed neighborhood about the (k, I)th neuron. First note that the LTA state \nequation simply increments the LTA state by one every time the neuron's STA state \ntoggles its output. Since a neuron oscillates after it has been initially activated, the \nLTA state, will represent the time at which the neuron was first activated. This \ntime, in turn, will simply be the \"length\" of the shortest path from the site of \nthe initial distrubance. In particular, consider the neighborhood set for the (k,l)th \nneuron and let's assume that the (k, I)th neuron has not yet been activated. If the \nneighbor has been activated, with an LTA state of a given value, then we see that \nthe (k,l)th neuron will be activated on the next cycle and we have \n\nWk,l = max \n(m,n)eN\"\" \n\n( \nwm,n -\n\nIIPk,,-pm,nlloo) \n\n~ \n\n(6) \n\nThis is simply a dual form of the Bellman iteration shown in equation 3. In other \nwords, over the free-space neurons, we can conclude that the network is solving the \nBellman equation with respect to the supremum norm. \n\nIn light of the preceding discussion, the use of cooperative neural fields for path \nplanning is straightforward. First apply a disturbance at the neuron mapping onto \nthe desired terminal position, P f and allow the field to generate STA oscillations. \nWhen the neuron mapping onto the robot's current position is activated, stop the \noscillatory behaviour. The resulting LTA state distribution for the (i, j)th neuron \nequals the negative of the minimum distance (with respect to the sup norm) from \nthat neuron to the initial disturbance. The optimal path is then generated by a \nsequence of controls which ascends the gradient of the LTA state distribution. \n\n\fOscillatory Neural Fields for Globally Optimal Path Planning \n\n543 \n\nfig 1. STA activity waves \n\nfig 2. LTA distribution \n\nSeveral simulations of the cooperative neural path planner have been implemented. \nThe most complex case studied by these simulations assumed an array of 100 by 100 \nneurons. Several obstacles of irregular shape and size were randomly distributed \nover the workspace. An initial disturbance was introduced at the desired terminal \nlocation and STA oscillations were observed. A snapshot of the neuronal outputs \nis shown in figure 1. This figure clearly shows wavefronts of neuronal activity prop(cid:173)\nagating away from the initial disturbance (neuron (70,10) in the upper right hand \ncorner of figure 1). The \"activity\" waves propagate around obstacles without any \nreflections. When the activity waves reach the neuron mapping onto the robot's \ncurrent position, the STA oscillations were turned off. The LTA distribution re(cid:173)\nsulting from this particular simulation run is shown in figure 2. In this figure, light \nregions denote areas of large LTA state and dark regions denote areas of small LTA \nstate. \n\nThe generation of the optimal path can be computed as the robot is moving towards \nits goal. Let the robot's current position be the (i,j)th neuron's position vector. \nThe robot will then generate a control which takes it to the position associated with \none of the (i,j)th neuron's neighbors. In particular, the control is chosen so that \nthe robot moves to the neuron whose LTA state is largest in the neighborhood set, \nNi,j' In other words, the next position vector to be chosen is Pk,l such that its LTA \nstate is \n\nWk I = max wz:,y \n\n, \n\n(z:,Y)EN i,j \n\n(7) \n\nBecause of the LTA distribution's optimality property, this local control strategy is \nguaranteed to generate the optimal path (with respect to the sup norm) connecting \nthe robot to its desired terminal position. It should be noted that the selection of \nthe control can also be done with an analog neural network. In this case, the LTA \n\n\f544 \n\nLemmon \n\nstates of neurons in the neighborhood set, Ni,j are used as inputs to a competitively \ninhibited neural net. The competitive interactions in this network will always select \nthe direction with the largest LTA state. \nSince neuronal dynamics are analog in nature, it is important to consider the impact \nof noise on the implementation. Analog systems will generally exhibit noise levels \nwith effective dynamic ranges being at most 6 to 8 bits. Noise can enter the network \nin several ways. The LTA state equation can have a noise term (LTA noise), so that \nthe LTA distribution may deviate from the optimal distribution. In our experiments, \nwe assumed that LTA noise was additive and white. Noise may also enter in the \nIn this case, the robot's next \nselection of the robot's controls (selection noise). \nposition is the position vector, Pk I such that Wk I = max( \n)EN . . (wx y + Vx y) \nwhere Vx,y is an i.i.d array of stochastic processes. Simulation results reported \nbelow assume that the noise processes, Vx,y, are positive and uniformly distributed \ni.i.d. processes. The introduction of noise places constraints on the \"quality\" of \nindividual neurons, where quality is measured by the neuron's effective dynamic \nrange. Two sets of simulation experiments have been conducted to assess the neural \nfield's dynamic range requirements. In the following simulations, dynamic range is \ndefined by the equation -log2lvm I, where IVm I is the maximum value the noise \nprocess can take. The unit for this measure of dynamic range is \"bits\". \n\nX,1J \n\n) \n\n) \n\n1,1 \n\nI \n\n) \n\nThe first set of simulation experiments selected robotic controls in a noisy fashion. \nFigure 3 shows the paths generated by a simulation run where the signal to noise \nratio was 1 (0 bits). The results indicate that the impact of \"selection\" noise is \nto \"confuse\" the robot so it takes longer to find the desired terminal point. The \npath shown in figures 3 represents a random walk about the true optimal path. \nThe important thing to note about this example is that the system is capable of \ntolerating extremely large amounts of \"selection\" noise. \n\nThe second set of simulation experiments introduced LTA noise. These noise ex(cid:173)\nperiments had a detrimental effect on the robot's path planning abilities in that \nseveral spurious extremals were generated in the LTA distribution. The result of \nthe spurious extremals is to fool the robot into thinking it has reached its terminal \ndestination when in fact it has not. As noise levels increase, the number of spurious \nstates increase. Figure 4, shows how this increase varies with the neuron's effective \ndynamic range. The surprising thing about this result is that for neurons with as \nlittle as 3 bits of effective dynamic range the LTA distribution is free of spurious \nmaxima. Even with less than 3 bits of dynamic range, the performance degradation \nis not catastrophic. LTA noise may cause the robot to stop early; but upon stop(cid:173)\nping the robot is closer to the desired terminal state. Therefore, the path planning \nmodule can be easily run again and because the robot is closer to its goal there will \nbe a greater probability of success in the second trial. \n\n4 Extensions and Conclusions \n\nThis paper reported on the use of oscillatory neural networks to solve path plan(cid:173)\nning problems. It was shown that the proposed neural field can compute dynamic \nprogramming solutions to path planning problems with respect to the supremeum \nnorm. Simulation experiments showed that this approach exhibited low sensitivity \n\n\fOscillatory Neural Fields for Globally Optimal Path Planning \n\n545 \n\n~~---r----.---~----'----' \nN \na \na \nN \n\n(/) \n(1) \n\n(/) \n\nC6 \nU5 \n::l o \n::l a. en \n15 \n\n.~ \n\n~ \n(1) \n.0 \nE \n::l \nZ \n\nDynamic Range (bits) \n\no \n\n1 \n\n2 \n\n3 \n\n4 \n\nfig 4. Dynamic Range \n\nc. ... \n\nfig 3. Selected Path \n\nto noise, thereby supporting the feasibility of analog VLSI implementations. \n\nThe work reported here is related to resistive grid approaches for solving optimiza(cid:173)\ntion problems (Chua, 1984). Resistive grid approaches may be viewed as \"passive\" \nrelaxation methods, while the oscillatory neural field is an \"active\" approach. The \nprimary virtue of the \"active\" approach lies in the network's potential to control the \noptimization criterion by selecting the interconnections and rate constants. In this \npaper and (Lemmon, 1991a), lateral interconnections were chosen to induce STA \nstate oscillations and this choice yields a network which solves the Bellman equation \nwith respect to the supremum norm. A slight modification of this model is currently \nunder investigation in which the neuron's dynamics directly realize the iteration of \nequation 6 with respect to more general path metrics. This analog network is based \non an SIMD approach originally proposed in (Lemmon, 1991). Results for this field \nare shown in figures 5 and 6. These figures show paths determined by networks \nutilizing different path metrics. In figure 5, the network penalizes movement in all \ndirections equally. In figure 6, there is a strong penalty for horizontal or vertical \nmovements. As a result of these penalties (which are implemented directly in the \ninterconnection constants D1:1), the two networks' \"optimal\" paths are different. \nThe path in figure 6 shows a clear preference for making diagonal rather than verti(cid:173)\ncalor horizontal moves. These results clearly demonstrate the ability of an \"active\" \nneural field to solve path planning problems with respect to general path metrics. \nThese different path metrics, of course, represent constraints on the system's path \nplanning capabilities and as a result suggest that \"active\" networks may provide a \nsystematic way of incorporating holonomic and nonholonomic constraints into the \npath planning process. \n\nA final comment must be made on the apparent complexity of this approach. \n\n\f546 \n\nLemmon \n\nfig 5. No Direction Favored \n\nClearly, if this method is to be of practical significance, it must be extended beyond \nthe 2-DOF problem to arbitrary task domains. This extension, however, is nontriv(cid:173)\nial due to the \"curse of dimensionality\" experienced by straightforward applications \nof dynamic programming. An important area of future research therefore addresses \nthe decomposition of real-world tasks into smaller sub tasks which are amenable to \nthe solution methodology proposed in this paper. \n\nAcknowledgements \n\nI would like to acknowledge the partial financial support of the National Science \nFoundation, grant number NSF-IRI-91-09298. \n\nReferences \n\nS.H. Benton Jr., (1977) The Hamilton-Jacobi equation: A Global Approach. Aca(cid:173)\ndemic Press. \n\nA.E. Bryson and Y.C. Ho, (1975) Applied Optimal Control, Hemisphere Publishing. \nWashington D.C. \nL.O. Chua and G.N. Lin, (1984) Nonlinear programming without computation, \nIEEE Trans. Circuits Syst., CAS-31:182-188 \nM.D. Lemmon, (1991) Real time optimal path planning using a distributed comput(cid:173)\ning paradigm, Proceedings of the Americal Control Conference, Boston, MA, June \n1991. \nM.D. Lemmon, (1991a) 2-Degree-of-Freedom Robot Path Planning using Coopera(cid:173)\ntive Neural Fields. Neural Computation 3(3):350-362. \n\n\f", "award": [], "sourceid": 571, "authors": [{"given_name": "Michael", "family_name": "Lemmon", "institution": null}]}