# State propagation from uncertain control input

Robotics Asked by Astghik Hakobyan on October 3, 2021

Consider a nonlinear system $$x(k+1)=f(x(k),u(k))$$, where $$x(k)inmathbb{R}^{n}$$ is the state, $$u(k)inmathbb{R}^m$$ is the control input. Here $$u(k)$$ is normally distributed RV with mean $$mu_u(k)$$ and variance $$Sigma_u(k)$$. I want to find the distribution of $$x(k)$$ starting from some deterministic $$x(0)$$ for some horizon $$K$$.
Then can the ordinary EKF update equation be used for the prediction in the below way?

begin{align*} mu_x(k+1)&=f(mu_x(k),mu_u(k)),\ Sigma_x(k+1)&=nabla_x f(mu_x(k),mu_u(k))Sigma_x(k)nabla_x f(mu_x(k),mu_u(k))^top+nabla_u f(mu_x(k),mu_u(k))Sigma_u(k)nabla_u f(mu_x(k),mu_u(k))^top end{align*}

Yes. This approach is commonly used when IMUs are present because they measure the rate of state change with some uncertainty.

Answered by holmeski on October 3, 2021

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