Degenerate Problem: State space model

Now we investigate the conditions for the intelligent decision problem of the firefighting robot to degenerate to the traditional state space estimation problem.

When the mapping Ψ: is an identical mapping, we are able to estimate the event parameter by estimating the state through (3.7), which will be proved in the following. And we can formulate the state space model based on observation model (3.7). Consequently, to degenerate this problem to the traditional form, we must find out the condition for Ψ to be an identical mapping.

We start by investigating , which generates the mapping Ψ. When we have precise knowledge of and robot’s location x, the relationship between state s (gradient of the potential field) and event parameter should be a deterministic function:

, 3.8 , can be determined by . However, we assume unknown and no available location information. Consequently, we model their relationship by conditional probability p | according to Proposition 2-4. However, the mapping reduces to an identical mapping under appropriate conditions. If we know satisfies the following condition,

c ,

for all , and is an arbitrary scalar function of 3.9 we have

p Arg Arg | 1 (3.10) (3-10) is equivalent to p | 1 when we consider the expected a posterior utility E cos Arg Arg | . There are many kinds of potential field

satisfying this condition, including electrostatic field and diffusion in free space [6].

As long as the temperature distribution could be modeled by these kinds of potential fields, this condition holds. Then following the concept of Corollary 2-5, we can formally prove that the optimal decision mapping can be constructed by estimating state under condition (3.9).

Corollary 3-1: Estimating state (gradient of potential field) is equivalent to estimating event parameter in the sense of utility function (3.1) if (3.9) (or equivalent, (3.10)) holds.

Proof:

We need to prove that under the condition (3.10), we have arg max

′ E cos Arg ′ Arg | arg max E cos Arg Arg |

As mentioned above, p | 1 is equivalent to (3.10) when calculating E cos Arg Arg | . Hence we use p | 1 instead. The maximum expectation of utility function is

max E u ,

max E cos Arg Arg | 3.12

max cos Arg Arg p | p | p d d 3.13 max cos Arg Arg p | p d 3.14 The equality of (3.13) and (3.14) holds because p | 1 and thus

p | 0 Hence

arg max cos Arg Arg p | p d 3.15 according to (3.11). Then the Corollary is proved. Q.E.D

Corollary 3.1 shows that under the condition (3.9) on the potential field, the observed physical quantity is identical to the event parameter for the utility function (3.1). Consequently, we are able to use the linear approximation observation model (3.7) to estimate the physical quantity (state) to make action decision without considering event parameter.

In addition to the observation model, we proceed to derive the state transition and formulate the problem into the state space estimation problem. We observe that the gradient is always point to wherever the robot stands on. Hence it is always the same when the robot makes right direction decisions, and would not change significantly even when it makes wrong direction decisions due to the small displacement between two observation collections assumed in the observation model.

Recall that the state is defined to be the direction of the destination in robot’s own coordinate system and this relative coordinate would rotate when the robot turns its direction, or say makes wrong decision. Hence change of the gradient’s direction comes mostly from the relative coordinate rotation instead of the gradient’s direction rotation in absolute coordination system when the robot makes wrong decision. We can conclude from above observations that the direction of gradient in the robot’s own coordinate system is the difference between in the direction of estimated gradient and the true gradient. Then the state transition is:

3.16

u is a random variable accounts for the direction deviation of the robot’s movement due to obstacles, mechanical errors or other non-ideal effects. Here we use normalized gradient to represent its direction. is the position vector of n’th observation.

The similar navigation problem has been investigated in [5] in a simplified version. We generalized the discrete direction selection solved by Maximum a posterior (MAP) hypothesis testing to the continuous vector form (3.17) and relieve the unrealistic assumptions on the observation model to derive the linear approximate observation model (3.7). Then the firefighting robot navigation problem can be formulated by modified state-space model with the estimation of previous state in state transition:

Then we can adapt the widely used MMSE method, Kalman filter, to solve it. In order to apply Kalman filter, we should make a further assumption to limit the error is in a small range in which the approximation

cos arg arg ~1 | | 3.18 holds. The assumption assures Kalman filter to fit the optimal decision mapping to maximize the utility function because it makes MMSE decision. However, (3.17.a) and (3.17.b) is different from traditional state-space model of Kalman filter problem due to the additional term, estimation of previous state, in (3.17.a). In fact, in many papers, for example [32], the estimation of previous state has been applied in the prediction problem. In [5], we solve this problem by directly applying MAP hypothesis testing due to the discrete direction form. But when we generalize it to

continuous direction, we should solve it by modifying solution of Kalman filter.

However, it turns out that the estimation of previous state can be regard as outside input in solving Kalman filter. We prove this in the Lemma A1 in Appendix 3. Then with this lemma, we can solve the state space model by Kalman filter approach. The solution is:

Prediction Phase

(3.19) (3.20) Estimation Phase

T T T (3.21)

(3.22) (3.23) Table. 3.2 Kalman filter

where

E ^T , E ^T

We can use above equations to recursively solve our state space model estimation problem.

Appendix 3.A State-space Model with Estimation of