Section 2

What is the definition of a non-holonomic constraint?

A non-holonomic constraint is one which limits the allowable velocities of the system (for example, the constraint which prevents cars from accelerating sideways)

When is a dynamical system called symmetric?

A system is symmetric if it is drift-free and the control set is symmetric about the origin. If the system has drift, it is symmetric if there is a linear combination of controls that cancels the drift.

What is the difference between the Dubins car and the Reeds Shepp car models? What is the impact on the definition of shortest paths for the two systems.

Dubins can go forward at constant speed, Reed-Shepp can move forwards/backwards. In terms of shortest paths, there are more sequences available for the Reed-Shepp cars, and they can also navigate in sequences of several circular movements faster since they can reverse, instead of having to go around a larger portion of the circle like a Dubins car does.

How does the expression for the predicted mean $\bar{\mu_t}$ change in the Extended Kalman filter relative to the basic Kalman filter?

EKF: $\bar{\mu_t} = g(u_t, \mu_{t-1})$, as opposed to $A_t\mu_{t-1} + B_tu_t$

What is the nonlinear optimization formulation of direct trajectory planning?

Establish the objective function, the dynamic, obstacle/join limit, and actuator limit constraints, and the boundary conditions. Then, we use numerical methods to approximate the answer.

What is Shiller's Dubowski's time-optimal trajectory planning approach (TOTP)?

First, discretize the space (grid/roadmap), then define all possible collision-free paths between the start and goal. Next, compute rough lower-bound estimates for traveling times on these paths (maximum velocity limit). Select the fastest available path by smoothing, running the time-scaling algorithm to generate a cost, prune all candidates with a lower bound less than this cost, and come up with better estimates for the remaining paths.

What is the minimum number of probabilities we need to know if we want to compute whether the probability P(x|y) is greater than or lower than P(¬x|y) for two binary random variables X and Y?

For Bayes Rule for $P(x|y)$ needs $P(y|x)$, $P(x)$, $P(y)$, $P(\lnot x|y)$ needs $P(y|\lnot x), P(\lnot x)$, $P(y)$, so 5 in total.

What is the model of a simple car? How do the unicycle, Reeds Shepp and Dubins car models arise based on it?

For a simple car, the equations for motion are given by $x' = u_s\cos\theta$, $y' = u_s\sin\theta$, and $\theta' = \frac{u_s}{L}\tan u_\phi$. From here, the unicycle's control space is $U = [-1,1]\times [-\frac{\pi}{2}, \frac{\pi}{2}]$. For the Reeds-Shepp car, we have $u_s \in\{-1,0,1\}$, meaning that it can go forward, backwards, or stay at rest. For the Dubin's car, we have $u_s\in \{0,1\}$, meaning that it can only move forward or stay at rest. These models come from defining the control space for the bodies.

What is the model of a differential drive robot? How do shortest paths for differential drive robots look like?

For the equations of motion, we have $x' = \frac{r}{2}(u_l + u_r)\cos\theta$, $y' = \frac{r}{2}(u_l + u_r)\sin\theta$, and $\theta' = \frac{r}{L}(u_r - u_l)$. Here, $u_l$ and $u_r$ are the controls for moving each wheel, $r$ is the radius of the wheels, and $L$ is the distance between the center of the two wheels in terms of width. When taking the shortest path, the robot turns to face the goal position without translating, and then translating to the goal position. The shortest path will just be a series of rotations and then following a straight line.

When is a dynamical system accessible?

A system is accessible from a point $x$ if $R^M(x,\leq T)$ contains a full $n$-dimensional subset of $M$ for some $T > 0$, where $R^V(x,T)$ is the set of reachable points at time $T$ by trajectories inside of $V$, where $V$ is a neighborhood of $x$, and $R^V(x, \leq T) = \bigcup\limits_{0 < t\leq T}R^V(x,t)$. That is, accessible from $x$ if, for some $T > 0$< the union of all reachable points in time less than or equal to $T$ by trajectories inside of the state space contains an $n$-dimensional subset of the state space.

What is the idea behind differential flatness and how it can impact the motion planning problem?

A system is differentially flat if, given $n$ state variables and $m$ control inputs, there exist $m$ functions $y_i(x,u,u',u'',\dots, u^{(r)})$ such that the states and controls can be expressed by the $y_i$. These functions are known as flat outputs. This simplifies the motion planning problem by making it so that finding a feasible trajectory from $x(0)$ to $x(t_f)$ is just finding a curve $y(t)$ that satisfies constraints on $y(0), y'(0),\dots, y^{(p)}(0)$ and $y(t_f),y'(t_f),\dots,y^{(p)}(t_f)$.

Consider two random variables $X_1$ ∼ N($\mu_1,\Sigma_1$) and X2 ∼ N($\mu_2$,$\Sigma_2$). What distribution does the product p($X_1$) · p($X_2$) follow?

$N((\Sigma_1 + \Sigma_2)^{-1}(\Sigma_2\mu_1 + \Sigma_1\mu_2), (\Sigma_1^{-1} + \Sigma_2^{-1})^{-1})$

Consider a random variable X ∼ N(μ,$\Sigma$), which follows a Gaussian distribution with a mean vector μ and a covariance matrix $\Sigma$. If the random variable Y arises from the linear transformation Y = A · X + B, then what distribution does the random variable Y follow?

$N(A\mu + B, A\Sigma A^T)$

Consider two random variables $X_1$ ∼ N($\mu_1,\Sigma_1$) and X2 ∼ N($\mu_2$,$\Sigma_2$). What distribution does their convolution follow?

$N(\mu_1 + \mu_2, \Sigma_1 + \Sigma_2)$

{What is the probability P(x, y|z) if X and Y are conditionally independent? What is P(x|z)?

$P(x,y |z) = P(x|z)P(y|z)$. $P(x|z) = P(x|z,y)$

You may be asked about the relative relationship between the controllability, accessibility, STLA, STLC properties, i.e., which property implies the other and under which conditions.

$STLA\implies$ Accessible, Accessible $\implies STLA$ if all vector fields are analytic, $STLA\implies STLC$ if symmetric, $STLC\implies$ Controllable if manifold is connected and STLC everywhere.

What is the predictive belief computed by the Kalman filter given a prior belief of $x_{t-1}$ ~ $N(\mu_{t-1}, \Sigma_{t-1})$ and a linear dynamics model $p(x_t|u_t,x_{t-1}) = N(x_t;A_t\cdot x_{t-1} + B_t\cdot u_t, R_t)$?

$\bar{bel(x_t)} = \bar{\mu_t} = A_t\mu_{t-1} + B_tu_t $ ; $\bar{\Sigma_t} = A_t\Sigma_{t-1}A_t^T + R_t$

Why is the unicycle able to reach every state in its state space? Why is the PBWT able to reach every state in its state space?

For the unicycle, the lie bracket is linearly independent from the existing vector fields, which means that the dimension of the distribution is 3 at all states (regular distribution). For the PBWT, as long as the value $d\neq 0$, the dimension of the distribution is 6 at all states (non-regular case).

What are the benefits of using a Gaussian distribution to represent a belief distribution?

Gaussian distributions are easily parameterizable, operate in a continuous space, and allows us to use Kalman filters, which come up with optimal state estimates in presence of noise

What is the choice in grid-based Markov localization?

Get clarification on this tomorrow

How does the expression for the updated mean μt change in the Extended Kalman filter relative to the basic Kalman filter?

In the extended kalman filter, the updated mean is $\mu_t = \bar{\mu_t} + K_t(z_t - h(\bar{\mu_t}))$ as opposed to $\bar{\mu_t} + K_t(z_t - C_t\bar{\mu_t})$

How should we implement the idea of adding additional samples for dealing with particle deprivation?

Monitor the probability of sensor measurements, and inject random particles when the average estimate is lower than typical values for reasonable estimates.

What are the two main limitations of Kalman filtering?

Most systems aren't linear, bad approximations for multi-modal distributions

What is the set of available methods for solving direct trajectory planning?

Nonlinear optimization, non-convex optimization, sequential convex optimization, kinodynamic searches, and BestNear selection.

What is the objective of Bayesian filtering?

Objective of Bayesian filtering is to approximate a robot's current state given a stream of data observations $z$ and action data $u$ $d_t = \{u_1,z_1,\dots,u_t,z_t\}$, sensor model $P(z|x)$, action model $P(x|u,x')$, and prior probability of the system state $P(x)$.

What is the difference between passive and active sensors?

Passive sensors take information in to sense it, while active sensors emit their own energy and analyze the feedback

What are the axioms of probability theory? Show that Pr(¬A) = 1 − Pr(A) given the axioms.

Pr$(A\lor \lnot A) = P(A) + P(\lnot A) - P(A\land \not A) \implies \\ P(True) = P(A) + P(\lnot A) - P(False) \\ \implies 1 = P(A) + P(\lnot A) - 0 \\ \implies P(\lnot A) = 1 - P(A)$.

What is the difference between proprioceptive and exteroceptive sensors? Is a compass a proprioceptive or an exteroceptive sensor?

Proprioceptive are internal and measure information about the robot, exteroceptive are external and measure the environment. A compass is exteroceptive.

What are possible ways for drawing sampling from a probability distribution?

Rejection sampling: Given $f(x) < 1$ for all $x$, sample $x$ from a uniform distribution, sample $c$ from [0,1], and keep all samples $x$ such that $f(x) > c$ ; Importance Sampling Principle: Use a distribution $g$ to generate samples from $f$, introduce importance weight $w = f/g$ f-target, g-proposal, $f(x)>0\implies g(x)>0$

What is the purpose of odometry/wheel encoders? What are their advantages and disadvantages?

They are used to measure wheel position and rotation speed. They are simple and widely available, but they have limited speed and slippage is possible.

What is the purpose of tactile sensors?

They measure contact

What is the purpose of range sensors? Provide a couple examples of the underlying technology and how it works.

They measure distances to objects using light/sound waves. Examples are sonar, laser range finder, etc. They send out a wave and measure the reaction that comes back

What is the controllability question for a dynamical system?

This asks whether or not we can access all possible states with the given controls.

What is the objective of global, time-optimal, decoupled trajectory planning based on time-scaling?

To find the shortest duration trajectory between two states in a computationally efficient manner using a collision-free path planner and the time-scaling algorithm.

What is the problem of triangulation? What is the problem of trilateration? How do they differ?

Triangulation measures distance through angle measurement, while trilateration measures distance through distance measurement.

What is the probability of $P(x|z_1, . . . , z_n)$given the Markov assumption?

Under the Markov assumption, $z_n$ is conditionally independent of $z_1,\dots, z_{n -1}$ once we know $x$. Then, $P(x|z_1,\dots,z_n) = \eta P(z_n|x)P(x|z_1,\dots,z_{n-1})$

What is the approach of the Extended Kalman filtering for dealing with non-linear dynamics and observation models?

Use a first-order Taylor series expansion to approximate the system into a linear system

What does Quang-Cunog Pham's extension over the TOTP achieve?

We can apply it directly in the state space through an RRT approach (no quasi-static assumption needed)

How can we visualize the set of feasible time scaling accelerations at each point of the phase space representation?

We construct the velocity limit curve to denote where admissible paths can go.

How does discrete search work for a simple car model?

We define a set (finite) of actions for the car. Then, initialize a tree and list the open set with the starting configuration. While we still have elements in the open set, select the first one and remove. If this is in the goal region, we are done. Otherwise, if this configuration is not near a previously occupied configuration, we mark it as occupied, and, for all of our actions, integrate forward a fixed time to our new configuration. If the path between these two is collision free, we mark this new spot as a successor to the previous in the tree. Compute the cost, and place this new configuration in the open set, sorted by cost. Repeat until the open set is empty.

What is the idea in sequential quadratic programming?

We define a trust region as a box around the current point, and take the second order Taylor expansion. Then, $h(x) = A(x^k)x + b(x^k)$. This converges to a local optimum, but quality is heavily dependent on initial guess. General usage is recommended as a smoothing technique once we have found a good initial guess.

What is a popular approach in the context of Kalman filtering for dealing with the fact that frequently the true underlying belief distributions are multi-modal? What is the issue that arises in this context?

We end up with multiple Kalman filters, one for each hypothesis. We have issues with corresponding certain observations to certain hypotheses, and we don't know when to delete/add a hypothesis. We extract the hypotheses from the sensing data and apply Bayes' rule to delete ones with low probability

What is the dimensionality of the state space of a simple car carrying $k$ trailers?

We need $x'$, $y',$ and $\theta'$ to define the coordinates and orientation of the car, and then $\theta_i$ to define the orientation of the $i$-th trailer. Hence, the overall dimensionality is $k + 3$, since there are $k$ trailers, each with an orientation angle, and 3 coordinates for the car.

What is the updated belief computed by the Kalman filter givne the predicted belief $\bar{x_t}$ ~ $N(x_t; \bar{\mu_t}, \bar{\Sigma_t})$ and the linear observation model $p(z_t|x_t) = N(z_t;C_t\cdot x_t, Q_t)$. Assume the expression for the Kalman gain $K_t = \bar{\Sigma_t}\cdot C_t^T\cdot (C_t\cdot \bar{\Sigma_t}\cdot C_t^T + Q_t)^{-1}$ known. Explain the expression for the new mean vector $\mu$ and what is the role of the Kalman gain in this expression.

$bel(x_t) = \mu(t) = \bar{\mu_t} + K_t(z_t - C_t\bar{\mu_t})$ ; $\Sigma_t = (I - K_tC_t)\bar{\Sigma_t}$ The Kalman gain details to use how much we want to change our current estimate.

If the robot's state space is n-dimensional, how many numbers do we need to keep track of in order to represent a Gaussian distribution in the robot's belief space.

$n^2$? I think? it says for $x$ and $y$ components we need an $n\times n$ matrix.

Describe the second order version of a unicycle, car, and/or differential drive.

$x' = s\cos\theta$, $y' = s\sin\theta$, $\theta' = u_\omega$, $s' = u_a$. \underline{Continuous-steering car}: $x' = \cos\theta$, $y'=\sin\theta$, $\theta' = \frac{\tan\varphi}{L}$, $\varphi' = \omega$, $\omega' = u_\alpha$, \underline{Differential drive}: $x' = \frac{r}{2}(\omega_l + \omega_r)\cos\theta$, $y' = \frac{r}{2}(\omega_l + \omega_r)\sin\theta$, $\theta' = \frac{r}{L}(\omega_r - \omega_l)$, $\omega_l' = u_l$, $\omega_r' = u_r$.

What are the flat outputs for the PBWT and what do they physically correspond to?

$y_1 = x_1 + \frac{1}{d}\cos x_3$, and $y_2 = x_2 + \frac{1}{d}\sin x_3$. This defines a point which is fixed relative to the body of the PBWT.

Discuss the effects of the choice of how to represent the belief distribution in the context of Bayesian filtering and robot localization.

*Occupancy grid* (Discretized grid-based ) grid-based Markov localization, simple and quick to update, but has resolution issues. *Gaussian distribution* (Easily parameterizable probability distribution): Uses an extended Kalman filter, operates in a continuous space, but has bad approximation for multi-modal distributions. *Particle Filter* (Represent distribution as a population of estimates): uses a particle filter, can adapt to complex distributions, but increasing accuracy requires a lot of computation.

What is the idea in particle filtering for representing belief distributions?

Distributions are represented as a population of estimates (random samples)

What is an Inertial Measurement Unit (IMU)? What are its two components and what does it measure?

An IMU uses gyroscopes and accelerometers to estimate the relative positions and six degrees of velocities (translation/rotation).

What is the definition of an underactuated system?

An underactuated system is one in which the number of controls is less than the number of dimensions in the state space. For example, a car is underactuated because we can turn the car and accelerate forwards and backwards, but we cant move perpendicularly to the direction of the wheels. Here, the state space is 5 dimensional ($x,y,\theta,\omega,\eta)$, while the control space is only two dimensional $(\omega', \eta')$

Describe at a high-level how shortest paths for the Dubins car look like

Any shortest path will be a combination of rotations of a certain arc around a circle, and could possible include straight-line translations.

What does the Bayes formula specify? How does it arise from the definition of conditional probability? What is the normalization-based version of the formula? What if there is background information available?

Bayes Formula states that $P(x|y) = \frac{P(y|x)P(x)}{P(y)}$, or that P(prior given the evidence) is $\frac{\textrm{P(likelihood)}\cdot \textrm{ P(prior)}}{\textrm{P(evidence)}}$. It arises from $P(x,y) = P(x|y)P(y) = P(y|x)P(x)$. The normalization based formula is $P(x|y) = \eta P(y|x)P(x)$. If we have background information $z$, then we have $P(x|y,z) = \frac{P(y|x,z)P(x|z)}{P(y|z)}$

What are some of the computational challenges of grid-based Markov localization and how can we avoid them in practice?

Computational issues arise with resolution leading to bad paths if too low, but discretization at too high of a level is computationally hard. We can fix this by pruning over certain grid cells that have a low probability to prevent from having to iterate over them all, and assume that the transition model is bounded.

What is the definition of conditional probability? What is the probability P(x|y) if the random variables X and Y are independent?

Conditional independence: if $x$ and $y$ are conditionally independent, then $P(x,y|z) = P(x|z)P(y|z)$. If $x$ and $y$ are independent, then $P(x|y) = P(x)$.

When should we prefer to use decoupled trajectory planning and when should we prefer to use direct trajectory planning?

Decoupled is preferable for lower-dimensional spaces, but with more complex motion constraints we would prefer kinodynamic.

What is the objective of direct trajectory planning?

Direct trajectory planning finds the optimal trajectory by computing the optimal path directly in the complete state space, instead of finding collision-free paths and then performing time-scaling.

You may be provided a small example for a robot operating in a two-dimensional, grid-base world and asked to compute the probability of where it is after a sequence of actions and observations.

Get the odometry $o_t$, and consider all cases for where the robot could've come from to get to a spot. At each spot, we have $P(x_{i,t}|x_{j,t-1},o_t) = P(x_t = gridcell | x_{t-1} prev_gridcell, o_t = o_t))\cdot P(x_{t-1} = prev_gridcell)$ The problem will include grids showing the probability of moving from one square to another, so you use those to determine $P(x_{i,t}|x_{j,t-1}, o_t)$ Then, from all of the cases, sum them up.

Provide a definition of decoupled trajectory planning.

Given a path $q:[0,1]\to Q$, an initial state $(0,s_0')$, $s_0' \geq 0$, and a final state $(1,s_f'), s_f'\geq 0$, find a monotonically increasing, twice-differentiable time scaling $s:[0,t_f]\to [0,1]$ such that $s(0) = 0, s(t_f) = 1$, $s(0)' = s_0'$, and $s_f'(t_f) = s_f'$. Also, this minimizes the total travel time $t_f$ while respecting the constraints for all time $t\in [0,t_f]$, $L(s,s') \leq s'' \leq U(s,s')$.

Provide a graphical representation of the Markov and stationarity assumptions in the context of robot localization, i.e., what is the dynamic Bayesian network that we consider in the context of robot localization?

Graphical representation is a graph, $x_{t-1}$, $x_t$, $x_{t + 1}$ DAG across the middle, at each $x_k$, the node for $u_{k-1}$ goes into $x_k$ from the top, and $x_k$ goes into $z_{k-1}$ from the bottom. We consider a static world, independent noise, a perfect model with no approximation errors.

What is Pr(x, y) if the random variables X and Y are independent?

If they are independent, $P(x,y) = P(x)P(y)$.

When is a dynamical system controllable?

If, given any state $x$, and any goal state $x_{goal}$, there exists a trajectory from $x$ to $x_{goal}$ in finite time.

What does the notion of bang-bang control mean?

In bang bang control, the selected acceleration is either $L(s,s')$ or $U(s,s')$ (max/min acceleration)

What is the importance of the Dubins (and Reeds Shepp) car model for motion planning of wheeled systems?

In both cases, each possible shortest path can be designated as one out of a finite number of sequences of finite length. In Dubins, they are one of 6 sequences of 3, and in Reed-Shepp, we have one out of 48 sequences of maximum length 5.

How can we compute the probability P(x|u) where x is the robot's state and u is the set of controls applied at a previous state x'?

In discrete case, $P(x|u) = \sum P(x|u,x')P(x')$, continuous: $P(x|u) = \int P(x|u,x')P(x')\diff x'$

What is a drift vector field?

In second order systems, the body will continue to move even if no controls are applied. In this case, the drift vector field visualizes how the body moves with 0 controls.

What does the law of total probability specify? What do its two implications specify, i.e., marginalization and conditioning?

In the discrete case, the law of total probability states that $P(x) = \sum\limits_yp(x|y)p(y)$, and in the continuous case $P(x) = \int p(x|y)p(y)\diff y$. This allows us to marginalize without having a direct probability distribution of a variable, say $X$. Instead, if we are given the joint distribution of $X$ and $Y$, we can use this law to marginalize $P(x)$. We then use this to find the value of, say $P(y)$ in a conditional $P(x|y)$ when applying Bayes' Formula.

What is the set of inadmissible regions in the phase space representation? What are inevitable inadmissible states in the same context?

Inadmissible regions in the phase space representation are those outside of the velocity limit curve. (no feasible acceleration) Inevitably inadmissible regions are those in which the integral curve from the state, with tangents inside the tangent cones, eventually reaches the inadmissible region.

How does sampling-based kinodynamic planning work?

Instead of sampling states for our roadmap, we sample controls. We select a node and then propagate a random control with a random duration. We also take Voronoi bias into consideration.

Describe particle filtering for robot localization according to the Sequential Importance Resampling (SIR) approach. Provide an algorithmic.

Interpret each weight $w_k^i$ as the probability of obtaining the sample index $i$ in the set $\{x_k^i : i = 1,...,N\}$. Draw $N$ samples from that discrete distribution and replace the old sample set with the new one. Set all weights to the constant value $\frac{1}{N}$

What are the desirable properties of the Kalman filtering and under what assumptions?

Kalman filters are efficient, if $k$ is the dimensionality of our measurements and $n$ is the dimensionality of our state, then $O(k^{2.367} + n^2)$, and if we have the assumption of Linear Gaussian systems, we have great optimality.

What is the objective of kinodynamic search and how did Canny, Donald, Reif and Xavier's approach work?

Kinodynamic search performs a direct search in the state-space. CDRX's approach discretizes the control set to $[-a_{max}, 0, a_{max}]$ (bang bang control), and discretizes the timestep by selecting a timestep $h$. We then do a BFS-like search by integrating controls forward, and prune nodes if the trajectory to it intersects obstacles or exceeds constraints. We terminate when we reach the goal state.

Provide the flat outputs of a unicycle and show how they can be used to motion plan for the system.

Let $y_1 = x_1$ and $y_2 = x_2$. Then, $[x_1,x_2, x_3]^T = [y_1,y_2,\tan^{-1}\frac{y_2'}{y_1'}]^T$, and $[u_1,u_2]^T = [\pm\sqrt{y_1'^2 + y_2;^2}, \frac{y_1'y_2'' - y_1''y_2'}{y_1'^2 + y_2'^2}]^T$. Then, for the constraints, we just plug in the values from $t = 0$ and $t = t_f$ and return what the corresponding $y_i$ is. For example, $y_1(0) = 0$, since in the initial state, $[0,0,0]$, $x_1 = 0$ and $y_1 = x$. However, $y_2(t_f) = 1$, since $x_2 = 1$ in the final state and $y_2 = x_2$.

What types of robotics problems arise in the context of partial observability and noise?

Localization (knowing where the robot is relative to the map), Robot Mapping (computationally effectively representing the world), SLAM (both), dealing with sensor inaccuracy

When is a dynamical system small-time locally controllable?

STLC from $x$ if $R^V(x,\leq T)$ contains a neighborhood for $x$ for all neighborhoods $V$ and all $T>0$.

What is the implication of a dynamical system being STLC in terms of motion planning?

STLC systems can maneuver through clutter: Any motion of a system without motion constraints can be approximated by a system that is STLC everywhere. (justifies the use of a decoupled approach).

Describe particle filtering for robot localization according to the Sequential Importance Sampling (SIS) approach.

Sample $N$ $x$'s from distribution space, initialize $N$ weight variables to 1. Iterate over $t = 1,2,..$ (Dynamics update): FOR each $i = 1,2,...,N$, sample $x_{t + 1}^i$ from $P(X_{t + 1}|X_t = x_t^i,u_{t+1})$ END FOR (Observation update): FOR $i = 1,2,...,N$, $w_{t + 1}^i = w_t^i\cdot P(z_{t+1}|X_{t+1} = x_{t+1}^i)$ ENDFOR Then, at any time $t$, the distribution is represented by the weighted set of samples $\{<x_t^i, w_t^i> i = 1..N\}$

How does the global positioning system work?

Satellites send orbital location plus time, and the receiver computes its location through trilateration and time correction.

What is a popular approach for solving non-convex optimization, which utilizes solvers for convex optimization problems? You may be asked to provide a graphical representation of the approach.

Sequential convex optimization is the method for this. See lecture notes for graphical representation.

When is a dynamical system small-time locally accessible?

Small-time locally accessible from a point $x$ if $R^V(x,\leq T)$ contains a full $n$-dimensional subset of $M$ for all neighborhoods $V$ and all $T > 0$.

What is the assumed expression by the (basic) Kalman filter in terms of how the system's underlying dynamics involve? What is the assumed expression by the (basic) Kalman filter in terms of how observations arise from the system's state?

State dynamics correspond to a linear function of state and control plus additive noise $x_t = A_tx_{t-1} + B_tu_t + \varepsilon_t$, $p(x_t|u_t, x_{t-1}) = N(x_t:A_tx_{t-1} + B_tu_t, R_t)$ Observations are also a linear function of the state plus the additive noise.

You may be given an example where a robot moves along recognizable feature and you may be asked to define the matrices that are needed in order to execute the Extended Kalman filter.

Take the Jacobian of $g$ with respect to the location ($G_t$), the Jacobian of $G$ with respect to the controls $(V_t)$, and Motion noise

You may be provided an example where a robot collects multiple measurements z1, z2, . . . , zn and we are interested in drawing samples from the distribution P($x$|z1, . . . , zn). You will be asked to show how we can define the weight of particles, if it is difficult to directly draw samples from this distribution but instead it is easier to draw from the individual distributions P($x$|z$_l$).

Target $f: p(x|z_1,z_2,\dots,z_n) = \frac{\prod_k p(z_k|x) p(x)}{p(z_1,z_2,\dots,z_n)}$, Sampling $g: p(x|z_l) = \frac{p(z_l|x)p(x)}{p(z_l)}$, importance weights is just $w = \frac{f}{g}$.

What is a Lie algebra of a set of vector fields G?

The Lie Algebra, denoted $\bar{Lie}(G)$, is the linear span of all Lie products, of all degrees, of vector fields in $G$. This turns the controllability question into whether $\bar{Lie}(G)$ of a system spans the whole state space, or just a subset of it. \\

What are Lie brackets?

The Lie Bracket of two vector fields in a distribution $D$ tells us whether infinitesimal motion along these motion vectors can generate motion not contained in their distribution. For example, in parallel parking, we cannot just slide in sideways to the parking spot. However, by repeatedly moving forward/backward and turning, we can get into the same state that sliding would give us. This implies that for the unicycle, which is underactuated in a 3D state space but 2D control space, we can reach the entire state space. \textit{Mathematical Expression}: Consider two vector fields $g_1$ and $g_2$. Consider that the robot starts at state $x_0 = x(0)$ and then executes the following actions: follows $g_1$ for a small amount of time $\varepsilon << 1$, $g_2$ for $\varepsilon$, $-g_1$ for $\varepsilon$, and finally $-g_2$ for $\varepsilon$. The composition of the infinitesimal motions is $x(4\varepsilon) = \varphi_\varepsilon^{-g_2}(\varphi_\varepsilon^{-g_1}(\varphi_\varepsilon^{g_2}(\varphi_\varepsilon^{g_1}(x_0))))$. Then, we take the Taylor series to get $x(4\varepsilon) = x_0 + \varepsilon^2(\frac{\partial g_2}{\partial x}g_1(x_0) - \frac{\partial g_1}{\partial x}g_2(x_0) + O(\varepsilon^3)$. The Lie Bracket corresponds to a new vector field, which is a new direction of motion that we can slowly move along. In the order of $\varepsilon^2$, we get $[g_1,g_2] = \frac{\partial g_2}{\partial x}g_1 - \frac{\partial g_1}{\partial x}g_2$.

What is the premise of the decoupled approach for trajectory planning?

The decoupled approach searches for the path in the configuration space first, and then finds a time-optimal scaling for that path subject to the actuator limits. On the other hand, the direct approach searches for a trajectory path that is simultaneously collision-free and time optimal.

What is the phase space representation?

The phase-space represents, at any given point $(s,s')$, we have a range of feasible accelerations $L(s,s') \leq s'' \leq U(s,s')$. We use this to find a curve from $(0,s_0')$ to $(1,s'_f)$ which is nonnegative everywhere, and the tangent at each state is inside of the cone at that state.

What are the issues with Sequential Importance Sampling?

The resulting samples are only weighted by themselves, and the samples themselves are never effected by the evidence. It fails to concentrate particles/computation in the high probability areas of the distribution.

What is the span(G) of a set of vector fields G?

The set of all linear combinations of vectors in $G$.

What is the particle deprivation issue? What approaches have been proposed as a solution?

While no information is obtained that favors one particle over another, due to resampling some particles with disappear and after running sufficiently long with very high probability all particles will have become identical which leads us to believe that the filter has determined the state. Resampling induces loss of diversity.Solution: Effective Sample Size - resample only when effective sampling size is low, Regularization- Consider that the particles represent a continuous density, and sample from that continuous density

Give an abstract algorithm for solving the Bayesian filtering problem.

\textbf{Bayes Filter(Bel(x), d):}\\ $\eta = 0$\\ $IF$ $d$ is a perceptual data item $z$ then\\ \indent $FOR$ all $x$ do\\ \indent\indent $Bel'(x) = P(z|x)Bel(x)$\\ \indent\indent $\eta = \eta + Bel'(x)$\\ \indent $FOR$ all $x$ do\\ \indent\indent $Bel'(x) = \eta^{-1}Bel'(x)$\\ $ELSE IF$ d is an action item $u$ then\\ \indent \textbf{FOR} all $x$ do\\ \indent\indent $Bel'(x) = \int P(x|u,x')Bel(x')\diff x'$\\ $RETURN$ $Bel'(x)$

Derive the Bayesian filtering expression. Indicate along the way what rules and assumptions you are applying to progress to the following step

\textbf{START}$Bel(x_t) = P(x_t|u_1,z_1,\dots,u_t,z_t)$\\ $\overrightarrow{BAYES} = \eta P(z_t|x_t,u_1,z_1,\dots,u_t)P(x_t|u_1,z_1,\dots,u_t)$\\ $\overrightarrow{MARKOV} = \eta P(z_t|x_t)P(x_t|u_1,z_1,\dots,u_t)$\\$\overrightarrow{TOTALPROB}=\eta P(z_t|x_t)\int P(x_t|u_1,z_1,\dots,u_t,x_{t-1}P(x_t|u_1,z_1,\dots,u_t)\diff x_{t-1}$\\ $\overrightarrow{MARKOV} = \eta P(z_t|x_t)\int P(x_t|u_tx_{t-1})P(t_{t-1}|u_1,z_1,\dots,u_t)\diff x_{t-1}$\\ $\overrightarrow{MARKOV} = \eta P(z_t|x_t)\int P(x_t|u_t,x_{t-1})P(x_{t-1}|u_1,z_1,\dots,z_{t-1})\diff x_{t-1}$\\ $= \eta P(z_t|x_t)\int P(x_t|u_t,x_{t-1})Bel(x_{t-1})\diff x_{t-1}$

You may be given a system (unicycle, PBWT, second-order car-like system, etc.) and asked to identify its configuration space, state space, control space, motion constraints, or its vector fields of motion.

\underline{Unicycle}: Configuration $[q_1,q_2,q_3]$, $q_1$ and $q_2$ are contact points on the plane, $q_3$ is the steering angle. The system is kinematic, $x = [x_1,x_2,x_3] = q$. Controls correspond to the rolling speed of the wheel and the rate of change of the steering angle. Non-holonomic constraint for prevent slipping while rolling: $x'\sin\theta - y'\sin\theta = 0$. \underline{Planar Body with Thrusters}: Body moves in a frictionless plane by means of two thrusters on the body, configuration $q = [q_1,q_2,q_3]$, $q_1$ and $q_2$ define the center of mass, $q_3$ defines the orientation. State space is 6 dimensional for coordinates and yaw/pitch/roll, $x = [x_1,x_2,x_3,x_4,x_5,x_6] = [q_1,q_2,q_3,q_4', q_5', q_6']$, control $u_1$ applies force/acceleration through the center of mass, while $u_2$ applies force perpendicular to the direction and at distance $d$ from center of mass, gravitational direction acts along $-q_2$.