Instructivo para la creación y conformación del Copasst Instructivo de Actuacion

In order to attenuate the state of a generic system (denoted as x(t)), the state it must be at least partially and preferably fully observable. These observations in robotics are usually obtained using sensors that have a degree of uncertainty in measurements. This makes it imperative to be able to model the statistics of these uncertainties in order better model the system and ultimately control the system. The multivariate probability density function will be denoted aspr(_·)which must satisfy

Z

pr(x)dx= 1 (3.99)

pr(x)expresses the likelihood thatxwill fall in the volumedx, hence integrating over the entire space the probability is1. The expectation operator is denoted as E[·], and for a function f(x) under a probability distributionpr(x)is defined as

E[f] =

Z

pr(x)f(x)dx (3.100)

wheref(x) =xis valid. The covariance is defined as

wherex,yare vectors andx = yis valid [13]. A covariance of an element xi or functionf(x) with itself is called the variance. A Gaussian distribution ofxdenoted as _N(x_|x¯,Σ)wherex¯,Σ are the mean and covariance matrix respectively, is defined as

N(x_|x¯,Σ) = _p 1 2π_|Σ_|exp −1 2(x−u) T_Σ−1_(x−u) (3.102)

The Gaussian distribution has the following propertiesR_−∞∞ N(x|x¯,Σ)dx= 1andN(x|x¯,Σ)>0. Unless explicitly stated or observed, it is generally assumed that uncertainty in most measurements of physically observed processes exhibit Gaussian noise about a mean. This is due to the Central Limit Theorem which states that for any distribution, as the number of samples approaches infinity the distribution converges to a normal distribution [13].

If it is assumed that the dynamical function or trajectory model has a unique output given partic- ular initial conditions, then when the initial conditions are known, previous states are not required to predict the processing of the state. This lays the foundation for state estimation using Markov processes, as a Markov process predicts a future state independent of past states given the current state of the system. A Hidden Markov Model (HMM) is a process where the actual statexis not observed and is referred to as the latent variable, however there is a projection to observed variables

z. The Kalman filter is one implementation of a HMM and a class of Bayesian filter, with a latent variable and assumes that all variables (observed and latent) have Gaussian distributions. There are a few extensions of the Kalman filter, two common ones are the Extended Kalman filter (EKF) and the Unscented Kalman filter (UKF), both of these improve on the approximation of the pro- cessing dynamics as shown in (3.95). These filters make the assumption that the state is measured discretely (which is the case using hardware), however the Kalman-Bucy filter provides a contin- uous time representation of the Kalman Filter. Due to its generality, the UKF is presented first, to handle non-linearity in the state-space the covariance is used to sample the local space based on the uncertainty using the following function

X=σ( ˆx,Σ) = ˆ x xˆ+γ√Σ xˆ−γ√Σ (3.103)

The estimated (or latent) variable in (3.103) isxˆwith associated uncertainty in the covariance matrix Σand the columns of the resulting matrix are referred to as the sigma points. For an-dimensional state space there are 2n+ 1sigma points in (3.103) defined by the columns of√Σ withγ as a constant variable defined as γ = α2_{(n+κ) where α, κ are adjustable, scalar parameters. The}

Kalman filter is described by an prediction step and an update step, for the UKF these are the prediction step: χ=f(χt−1,ut−1) ˆ xt=E[χt] Σt=cov[(χt−xˆ),(χt−xˆ)] +Rkf,t χt=σ( ˆxt,Σt) (3.104)

and for the update step:

Zkf =h(χt) ˆ zt=E[Zkf] Lkf,t=cov[(Zkf−zˆ),(Zkf −zˆ)] +Qkf,t Mkf,t=cov[(Xt−xˆt),(Zkf −zˆ)] Kkf,t=Mkf,tL−_kf,t1 ˆ xt= ˆxt+Kkf,t(zt−zˆt) Σt= Σt−Kkf,tLkf,tKkf,tT (3.105)

whereuis the control input,zis the measurement which is observed by the sensors, andRkf,t, Qkf,t are the uncertainties associated with the process and the measurement respectively [70]. The tradi- tional Kalman-Bucy filter was designed for linear time invariant systems, but many renditions exist including the hybrid Kalman-Bucy filter that has both continuous and discrete components, Ex- tended Kalman-Bucy filter, and finally the continuous time Unscented Kalman-Bucy filter (UKBF)

presented here for a system described by the dynamics and output

˙

x(t) =f(x(t),u(t), t) +Mkf(t)ηRkf

z(t) =h(x(t), t) +Lkf(t)ηQkf (3.106)

whereηQkf,ηRkf are white noises with spectral densitiesηQkf ∼ N(0, Qkf)andηRkf ∼ N(0, Rkf)

respectively and projection matricesLkf(t),Mkf(t). Then with a weighting matrix as a function of the weightingsγ: Wσ = Γkf(γ)for the sigma points as defined in [59], the complete continuous time UKBF can be represented as

dxˆ dt =f(X(t),u(t), t) +Kkf(Zkf −h(χ(t), t)) dΣ dt =X(t)Wσf T_{(χ(t),u(t), t) +f(χ(t),u(t), t)W} σX(t) +LkfQkfLTkf −KkfMkfRkfMkfT KkfT Kkf =X(t)WσhT(χ(t), t)Q−_kf1 (3.107)

as shown in [59, 86]. So these methods present a way to model the procession of states and the associated uncertainty given a valid dynamical model. However, if the estimating model is poor then the associated error can be expected to reflect the quality of modeling, in fact for drastically incorrect models, the white noise assertion in (3.106) becomes incorrect. Hence the challenge becomes improving the model representation during operation which will in turn improve estimation which will then in turn improve control. This motivates the need for model approximation, and that performed online in the form of adaptive control through model approximation.