All Sections
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 $y(t_f),y'(t_f),\dots,y^{(p)}(t_f)$.
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 are the properties of a motion planning roadmap?
A union of one-dimensional curves is a roadmap RM if for all q_{start} and q_{goal} in Q_{free} that can be connected by a path, the following 3 properties hold: 1) Accessibility: there exists a path from q_{start} within Q_{free} to some q_{goal} within RM 2) Departibility: there exists a path from some q_{goal} within RM to q_{goal} within Q_{free} 3) Connectivity: there exists a path in RM between q_{start} and q_{goal}
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)
What is a policy?
A policy $\pi$ is a function from a set of states $S$ to a set of actions $A$ which maximizes the expected discounted reward given an infinite horizon and assuming full observability.
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.
Provide the Bellman equation for MDPs
$V^*(s) = $ max$_{a\in Ap(s)}\sum\limits_{s'\in S}Pr(s'|s,a)[R(s,a,s') + \gamma V^*(s')]$
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 $ ; $\bar{\Sigma_t} = A_t\Sigma_{t-1}A_t^T + R_t$
What is a popular odometry model for a planar robot that can both rotate and translate?
$\delta_{trans} = \sqrt{(\bar{x}' - \bar{x})^2 + (\bar{y}' - \bar{y})^2}$ $\delta_{rot1} = atan2(\bar{y}' - \bar{y}, \bar{x}' - \bar{x}) - \bar{\theta}$ $\delta_{rot2} = \bar{\theta}' - \bar{\theta} - \delta_{rot1}$
How does the velocity-based motion model operate?
$\hat{\delta}_{rot1} = \delta_{rot1} + \varepsilon_{\alpha_1|\delta_{rot1}| + \alpha_2|\delta_{trans}|}$ $\hat{\delta}_{trans} = \delta_{trans} + \varepsilon_{\alpha_3|\delta_{trans} + \alpha_4|\delta_{rot1} + \delta_{rot2}|}$ $\hat{\delta}_{rot2} = \delta_{rot2} + \varepsilon_{\alpha_1|\delta_{rot2}| + \alpha_2|\delta_{trans}|}$
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.
Provide the equation expressing the probabilistic representation of SLAM. How does factoring this expressing into a robot path posterior and landmark position posterior help to approach the problem?
$p(x_{1:t}, l_{1:m} | z_{1:t}, u_{0:t-1}})$. Factoring into the path posterior and landmark posterior enables us to compute the landmark posterior efficiently, which allows us to use particle filtering.
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.
Identify 5 key modules of the PRM framework that need to be specified to have a concrete implementation of the approach
1) How to sample from the configuration space 2) How to connect configurations via a local planner 3) What is the right distance function (especially in a high-dimensional non-Euclidian configuration space 4) How to define the neighbors and find it efficiently 5) How to check for collisions along configuration paths
Provide the pseudo-code for the PRM
1) Take a random sample, determine whether a collision has occurred 2) Continue iteratively to achieve sufficient coverage of C_free 3) For pairs of neighboring configurations, evaluate whether a local path connecting them can be found in C_free. If yes, add an edge 4) Continue adding nodes and edges to (hopefully) determine the connectivity of C_free 5) Given start and goal configurations, connect to the nearest valid edge and discover path on map
Describe an example of a repulsive potential
2 like charges according to Coulomb's law
What is the notion of a homotopic class and how does it relate to the operation of local search methods for motion planning?
2 paths with the same endpoints are homotopic if they continuously deform in C_free
Why does the shortest path between 2 points among a set of disjoint polygonal obstacles consist of arcs of the visibility graph?
??? Same as the proof of the shortest path among set S of disjoint polygonal obstacles being the polygonal path whose inner vertices are vertices of S
What is the challenge with representing obstacles in high-dimensional configuration spaces where the robot is just a point?
??? The configuration space can be very complex? Hard to tell when point collides with obstacle?
How is a rigid body transformation in 2D represented?
A 2D rigid body transformation can be represented as a series of matrices. In particular, Final=T x Initial, where the final and initial are vectors (x, y, \theta)(P_1 = TxP_0) T = [cos \Delta \theta -sin \Delta \theta \Deltax sin \Delta \theta cos \Delta \theta \Delta y 0 0 1 ]
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.
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 is the difference between full SLAM and incremental SLAM?
Full SLAM estimates the entire path and map, incremental SLAM estimates the most recent pose and map.
What is the definition of a full-actuated (or underactuated) dynamical system?
Fully actuated: #controls=#DoF Underactuated: #controls < #DoF
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
What is the motivation for following a sampling-based approach for motion planning?
Reduces grid complexity; enables grid to be generalized to higher dimensions. Aims to relax the objective of completeness to probabilistic completeness.
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 does the notion of sample dispersion specify? How can it be employed in the context of sampling-based motion planning
Roughly means the largest ball/sphere that can be fit in the samples without including any sample inside the ball. Generally, given |P| samples, a sample set with smaller \delta(P) dispersion is better
What are the useful properties of quasi-random sampling processes for motion planning?
Runtime is guaranteed to be the same for all the runs due to the deterministic nature of the point generation process
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 $\{<x_t^i, w_t^i> i = 1..N\}$
Describe the key operations of the asymptotically optimal RRT* approach
Sample a nearby point, check for collision, check ~log(n) neighbors for connectivity
How do we sample uniformly in SE(2)
Sample all DoFs independently (as if R^3)
What were the choices of the original PRM implementation in terms of sampling/local planner/distance function/neighborhood selection and collision checking for rigid body motion planning?
Sampling: uniform sampling, deterministic sampling, quasi-random sampling local planner: Euclidian distance, find straight lines in rotation, spherical linear interpolation
What is the expression indicating the greedy policy choice in the context of RL? What is the problem with following the greedy policy?
GLIE, $\lim_{k\to \infty} N_l(s,a) = \infty$. The problem with the greedy policy is that we may put the robot in a hazardous position.
Show that "if G_2 is a node that corresponds to a goal state but a suboptimal path, A* will select a reachable node n along the optimal path before selecting G_2 for expansion"
G_2 will have a greater f-value than other nodes, so the other nodes will be expanded first before G_2. This will keep iterating and repeating and eventually G2 will be reached with the optimal path???
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
If there is no gradient information readily available, then provide an example of a local search approach that can still be used to identify candidate solution paths
Genetic algorithms: maintain a population of candidate solutions. Score in terms of validity and path quality. Selection: higher scoring paths have a higher probability of crossing over Crossover: share information among paths to generate new children Mutation: locally adapt resulting candidate solutions
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 the choice in grid-based Markov localization?
Get clarification on this tomorrow
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 (1) a path q: [0,1] -> Q (2) initial state (0, s._0), s._0 >= 0 (3) final state (1, s. _f), s._f >=0 Find monotonically increasing, twice-differentiable time scaling s:[0, t_f] -> [0,1] that: 1) Satisfies s(0)=0, s(t_f)=1, and s.(0)=s.(0), s.(t_f)=s._f 2) Minimizes total travel time along the path 3) While respecting the constraints for all time t in (0, t_f)
What does the Separating Axis Theorem specify and what is its usefulness in the context of collision checking?
Given 2 convex shapes, if there is an axis along which the projections of the two shapes do not overlap, the shapes do not collide. Finding this axis means there is no overlap between the 2 shapes and the query for collision checking can stop.
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 $t\in [0,t_f]$, $L(s,s') \leq s'' \leq U(s,s')$.
Specify three alternative methods that you are aware of for least squares optimization.
Gradient descent, Newton's Method, Guass-Newton
What is the form of classical search, which is useful for grid-based representations since it avoids repeated states? you need an additional data structure in order to be able to avoid repeated states. How is this data structure called and how can you implement it?
Graph Search. We need a closed list to keep track of searched nodes, since each node can only be expanded once. Typically implemented as a hash table.
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 are popular representation choices for maps?
Grid maps or scans, and landmark-based
What are popular representation choices for maps of 2D planar setups
Grid-based maps, polygonal obstacles in free space
Describe the key operations for Rapidly-Exploring Random Tree (RRT)
Grows from point, tries to connect new points to closest part of the existing tree. Check if sample falls inside obstacle for collision before connecting path
What desirable properties does A* exhibit for graph search under which conditions?
Guaranteed to find the shortest path with consistent heuristics
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.
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 $\bar{\mu_t} + K_t(z_t - C_t\bar{\mu_t})$
What is the set of inadmissible regions in the phase space representation? What are inevitable inadmissible states in the same context
Inadmissible regions are regions where the upper limit for acceleration is less than the lower limit (usually in the upper part of the phase space). Inevitably inadmissible states are when you're not quite in an inadmissible state yet, but it will be impossible to avoid reaching one.
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.
What is the idea behind the alternative scan-based model? You may be asked to draw an example.
Instead of following the path along the beam, simply check the endpoint.
What is the difference between model-free and model-based reinforcement learning?
Instead of learning the transition model and reward function explicitly, try to simultaneously compute the policy and the value function without learning the models.
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.
What is the key difference in the POMDP setup relative to MDPs given the fact that the state is not directly observable?
Instead of working with exact states, POMDPs make their own decisions based on the belief state.
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 is the benefit of using a KD-tree in the context of sampling-based motion planning?
It allows one to find the nearest neighbors of a point in O(log(n)) time
What is the purpose of the "verticle sweep-line" approach in the context of a trapezoidal decomposition
It reduces the computational time for the trapezoidal graph from O(n^2) to O(nlog(n))
What does the probabilistic completeness property specify? What types of motion planning problems is it applicable to?
It specifies that for any robustly feasible motion planning problem defined by P=(X_free, X_init, X_goal), the limit as n->infinity of the probability of the algorithm returning a solution is 1. This is applicable to motion planning problems with a robust solutions, Robust solution remains if a solutions if obstacles are dilated by some small ??? This dude didn't write the answer in grammatically correct English it's whatever
What is the principle difference between the EKF when it is applied to localization versus when it is applied to a SLAM problem?
It uses an extended vector state $y_t$ which comprises both the robot pose $x_t$ and the position of all features $m_i$ in the map ($y_t = [x_t,m_1,\dots,m_{n-1}]^T$
Using one or more of the criteria used to compare search algorithms, provide arguments to support or contest the following statements: 1) Iterative deepening is preferred over BFS 2) Bidirectional search is preferred over BFS
Iterative DFS: both have the same optimality, completeness, time complexity, but iterative DFS has better spatial complexity Bidirectional: They are both complete, neither is optimal unless weight nondecreasing, but spatial + temporal complexity of bidirectional search is O(b^d/2) whereas the complexities of BFS are O(b^d)
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.
What is the Lagrange function of a mechanical system
L = K - P where K is kinetic nergy and P is potential energy. KE = 0.5mv^2 + 0.5l\omega^2 (where v=dx/dt, v=dy/dt, \ometa=d\theta/dt) and potential energy = U = mgh
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
How can we change the reference frame of a robot?
Multiply by the transformation matrix of the new coordinate origin with respect to the previous
Provide the pseudo-code for RRT*
No thanks
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 does the property of asymptotic optimality specify? What types of motion planning problems is it applicable to?
An algorithm ALG is asymptotically optimal if for any motion planning problem P=(x_free, x_init, x_goal) and cost function c that admits a robust optimal solution with finite cost c*, P(lim i->infiinity of Y^ALG = c*) = 1 Applies to probabilistic motion planning problems with a robust optimal solution.
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 is the idea behind value iteration? Outline the algorithm. What properties does it exhibit?
Applies dynamic programming to refine approximations towards an optimal policy and its value. Iterations are computationally efficient, but convergence is asymptotic.
How can you achieve a probabilistically complete approach given a local search approach for path planning that has a positive probability of finding a solution?
Apply randomized restarting, essentially, you sample directly the space of candidate paths
What are the properties of BFS (or depth-first or uniform first or iterative deepening depth-first or bidirectional breadth-first) search? Justify the complexity bounds you report
BFS: Complete if b finite, Time O(b^d), Space O(b^d), not optimal unless cost is non-decreasing with depth. Uniform Cost: Complete if b finite and \eta > 0, Time O(b^{C*/\eta}), Space O(b^{C*/\eta}), Optimal DFS: Not complete, Time O(b^m), Space O(bm), Not optimal Iterative DFS: Complete if b is finite, Time O(b^m), Space O(bd), not optimal unless cost non-decreasing with depth Bidirectional BFS: Complete if b is finite, Time O(b^d/2), Space O(b^d/2), not optimal unless cost is non-decreasing with depth
What is the main idea behind potential functions?
By defining the obstacles as repulsive functions and the goal as an attractive function, the robot can follow the negative gradient of the potential function and approach the goal
How can multiple transformations of rigid bodies that rotate and translate in 2D (or 3D) can be composed?
Consecutive transformations can be chained together by multiplying matrices in series (rigid body transforms)
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.
Decoupled vs Coupled?
Decoupled: plan a path first, time scale later. Coupled: directly search phase space for path + time scaling
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.
What does the notion of \eta-goodness specify? What does the notion of \beta-lookout specify?
F=C_free is \eta-good if given any point q in F, \mu(V(q)) >= \eta \mu(F) where \mu measures the volume of a region, V is visibility. I.e. the volume that is visible from sample q is greater than \eta * total volume The \beta-lookout of a subset is the set of all configurations not in F2 that have line of sight to a fraction \beta of F2
Which optimal informed search algorithm expands fewer nodes than A* for the same heuristic? Ignore the effect of ties
FDA*
What is the definition of an admissible heuristic? WHat is the definition of a consistent heuristic?
Admissible: the heuristic cannot overestimate the optimal cost to the goal node from any point: h(x) <= C* Consistent: h(x) <= c(x, a, x') + h(x')
What are the advantaged and disadvantages of the visibility graph approach for computing paths for robots?
Advantages: it provides the shortest path between 2 obstacles, it only works over polygonal representation Disadvantages: shortest path can be too close to obstacles, it's computationally expensive, cannot be generalized to higher dimensions
What is the SLAM problem in robotics? Why is it a difficult problem?
Simultaneous Localization and Mapping. SLAM learns a map and locates the robot simultaneously. It's difficult because a map is needed for localizing a robot, and a pose estimate is needed to build the map.
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.
What is the issue that arises in cyclic environments when performing SLAM?
Small local errors accumulate to arbitrarily large global errors.
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 $T > 0$.
When two spaces are topologically equivalent? Why are we interested in the topology of configuration spaces?
Spaces are topologically equivalent when they are homeomorphic. That is, if there is a bijective (1:1 and onto) function from one space to the other.
What is the idea behind trapezoidal decomposition that allows it to be a complete path planning approach?
Split the free space into a series of trapezoids, build a roadmap based on neighboring shapes, connect the centers.
What are the principles of local search and how do they materialize in the context of motion planning?
Start from a complete but invalid candidate solution. Parametrize the solution in a way that can be updated via local adaptations (e.g. a curve with movable control points). Incrementally update the solution to improve its validity of quality (move away from obstacles to avoid collisions, once valid, tighten curve to shorten path)
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.
What is the Minkowski sum/difference?
Sum: expand obstacles (O) in the free space by the size of the robot (R) according to CO = O (+) R := {a + b | a \eta O, b \eta R} Difference: for a robot not symmetric about the origin (e.g. a car), flip the robot and take the minkowski sum to transform the obstacle space according to CO = O (-) R := {a - b | a \eta O, b \eta R}
Show that a consistent heuristic is also an admissible heuristic
Suppose a consistent heuristic h satisfies inequality h(x) <= c(x, a, x') + h(x'), where h(x_g)=0. Suppose the heuristic for a given node x is a value greater than C* by h. For a subsequent node to satisfy the triangle inequality h(x_{g-2}) <= c(x_g, a, x_{g-1}) + h(x_{g-1}), so C* + h <= (C* - C*) + h(x), and therefore h(x) >= C* + h. Because h is consistent, h(x_g)=0, so a node adjacent to the end node will encouunter the inequality h(x) <= C*. However, h(x) is greater than C* by at least h. This is a contradiction: a consistent heuristic must therefore be admissible. OR proof by induction.
What is the difference between Monte Carlo and Temporal Difference Learning? What is the update rule in TD learning?
TD can learn before knowing the final outcome and without knowing the final outcome.
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 the computational benefit of the Graph-based SLAM approach in terms of the emerging Jacobian matrix that arises in the corresponding least squares optimization process?
The Jacobian will be non-zero only in the rows corresponding to $x_i$ and $x_j$, which are the state variables which affect the error.
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.
How do we randomly sample uniformly in SE(3)
The frick if I know
What is the difference in focus between PRM and RRT? i.e. when should we use one versus the other
The main difference in focus between the 2 is the number of queries you expect to perform on the roadmap. For roadmap where you want to perform multiple queries, PRM is better option. If you want to perform single query, RRT is better because it's faster
What is the key representation choice in Graph-based SLAM for the landmark-based case? How does the representation change in the case of a grid-based representation?
The nodes of the graph represent robot locations and features in the map. Edges are relative coordinates of consecutive robot poses and relative positions between robot poses and features observed from the poses. For a grid based map, we use the graph and correct the nodes to render a map
What is the problem in using a grid-based representation in high-dimensional motion planning problems?
There are too many nodes to compute a solution effectively, as the spacial complexity of a side length n in $dimensions d increases as O(n^d)
What is the objective of the visibility graph?
There is a theorem stating that the fastest path through a field of polygonal obstacles intersects the vertices of the polygons within. By precomputing the visibility graph, the fastest path through the field can now quickly be computed with A*
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 are the properties of shortest paths in planar environments that contain disjoint polygonal obstacles?
They coincide with the vertices of the polygonal obstacles
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.
How can we compute the free subset of the configuration space for a (symmetric or asymmetric) polygonal robot and a polygonal obstacle in the environment?
This can be achieved by buffering the obstacles to treat the robot as a single point. This is accomplished using the Minkowski sum if symmetric, or Minkowski difference if asymmetric
What is the purpose of the rotational sweep approach in the context of the visibility graph
To decrease the runtime of creating the visibility graph to O(nlog(n))
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.
How does a KD-tree work?
Tree-like data structure for storing points in k dimensions. Construction: pick dimension i, pick a point with coordinates x=(x1, x2, ...xk), split points based on xi (greater or less), repeat recursively Depth log(n) if balanaced Construction takes O(knlog(n)) time, can speed up to O(nlog(n)) if balanced
What other types of convex decomposition do you know other than trapezoidal decomposition?
Triangular decomposition? Also possibly quad-tree adaptive decomp? oct-tree adaptive decomp???
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 challenge of feature correspondence? What is the benefit if we know the correspondences between features?
Trying to find the "important" points and to decrease the number of correspondences. Results in higher efficiency and higher accuracy.
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 information is typically stored on the nodes of a search tree for uninformed search? What additional info do we store in the case of informed search?
Uninformed: parents, neighbors. If Dijikstra: cost to node. Informed: parents, cost to node, heuristic to the goal.?????
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 is the main idea behind the Gauss-Newton method (no equations needed - but a qualitative description of what it optimizes) and what is its main issue?
Uses curvature information to take a more direct route for minimization. The problem is that the computation of the Hessian matrix is not trivial, and it might get stuck at a saddle point.
What is the representation issue with vertical cell decomposition in higher dimensions?
VCD requires the space of obstacles to be represented as polytopes or high-dimensional polygons. In the real world however, your configuration space for obstacles depending on the geometry of your robot can look rounded or circular
What is the extension of trapezoidal decomposition into higher dimensions? How does it work?
Vertical Cell Decomposition in higher dimensions: 1. Pick dimension x 2. Sweep the config space using a (n-1) dimensional plane orthogonal to x_i 3. Repeat the process to each of the (n-1) dimensional slices 4. The sweeping then yields convex n cells, (n-1) cells, and so on
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.
How does reinforcement learning relate to MDPs?
We don't know which states are good and what actions do, so we try actions and states out to learn.
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 data association problem in the context of SLAM?
When assigning observations to landmarks, we have more than $\binom{n}{m}$ possible associations, which is computationally difficult.
What does the notion of a "bang-bang control" solution mean
Whenever possible, fully saturate the controls (maximum acceleration/deacceleration)
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
When do sampling-based motion planning algorithms work well and when does their performance degrade?
Works reasonably in high-dimensional problems. Past 12 dimensions, the concept of Euclidean distance breaks down and these algorithms are no longer useful
Is the "Manhattan distance" an admissible heuristic when planning on a grid?
Yes, if and only if diagonal moves are prohibited, since if diagonal moves are allowed, it consistently overestimates the optimal distance to the end node, violating admissibility h(n) <= C*(n)
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 $-q_2$.
How does the general form of a dynamical system translate into an expression between controls and the time scaling parameter of a trajectory?
a(s)s.. + b(s)s.^2 + c(s) = u
You may be asked to provide the graphical model of online/incremental SLAM.
https://i.imgur.com/iSVxqT3.png
Describe the key operations of the Probabilistic Roadmap Method (PRM)
input: # of iterations n, configuration space Q, distance function. 1) How to sample 2) How to connect configurations (local planner) 3) What is the right distance function 4) How do we define the neighborhood and effectively find it (nearest-neighbors) 5) How to check collisions along configurations and paths
You may be asked to apply value iteration over a small scale example.
lmao pls no don't ask this
Is RRT asymptotically optimal
lol definitely not
What are the drawbacks of the scan-based model?
no explicit model of people and other dynamics that may cause short readings, no modeling of the beam (treats sensor as if it can see through walls), and can't handle unexplored areas
You may be asked to apply policy iteration over a small scale example.
pls no
You may be provided a small scale POMDP and asked to compute the policy for a short horizon.
pls no
Provide the pseudocode for the operation of a potential function
q = q_init While the gradient potential function is greater than 0: q = q - alpha * gradient where alpha is a step size doesn't need to be constant)
What is the objective of scan matching?
tries to incrementally align two scans or a map to a scan, without revising the past/map
What are the limits for the second-order derivative of the time scaling parameter in decoupled trajectory planning?
u^min(s, s. ) <= a(s)s.. + b(s)s. ^2 + c(s) <= u^max (s, s. )
What is the general form of dynamical systems that provides the relationship between controls and states?
u_i = M(q) * q^(2) + C(q, q^(1)) * q^(1) + g(q)
What is a "frame" in robotics?
A global frame is a worldwide fixed coordinate system. A local frame is a set of coordinates relative to the robot.
What is a navigation function and under what conditions can it be defined?
A navigation function is a potential function in a sphere based world with only one global minimum (no local minima). This is valid for any star shaped world that can be transformed continuously into a spherical representation
What property did the original PRM and RRT approaches provide?
A relaxed notion of completeness (probabilistic completeness). If there is a solution, these algorithms will find it with a probability of 1 as the number of samples grows greater and greater
What are semi-algebraic sets and what is their relevance to motion planning?
A semialgebraic set is a subset S of R^n defined by a finite sequence of polynomial equations and inequalities. These sets can describe the configuration space of the motion planning problem.
What are the elements of a Markov Devision Process? What is the output of an MDP?
A set of states $S$, a set of actions $A$, a transition model $Pr(s'|s,a)$, a reward model $R(s,a,s')$, a start state $s_0$, and a discount factor $\gamma$. The MDP finds a policy $\pi$.
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.
Show that "A* with a consistent heuristic finds an optimal path to all expanded nodes, namely g(n)=g*(n) for any expanded n"
Assume g(n)>g*(n) and n expanded along a non-optimal path. Let n' be the shallowest OPEN node on optimal path p to n Then g(n') = g*(n') and therefore f(n') = g*(n')+h(n') Due to consistency, we get f(n') <= g*(n') + c(n', n) + h(n) Since g*(n)=g*(n') + c(n', n) along the optimal path, we get that f(n') M= g*(n) + h(n) And since g(n) > g*(n), then f(n') < g(n) + h(n) = f(n). Contradiction.
Prove the lemma that states: "Any shortest path among a set S of disjoint polygonal obstacles is a polygonal path whose inner vertices are vertices of S"
Assume the shortest path T is not "polygonal. Then there is a point p on the path T that lies on the interior of the free space and no line segment containing p is in the path. So there must be a positive radius disc at p that is completely contained in the free space. Then the part of T inside the disc can be shortened by using a line segment connecting the point where it enters the disc to the point where it leaves it. This contradicts the optimality of T, so T is "polygonal"
What is the key assumption of the beam-based sensor model? What are the 4 components of the beam-based proximity model and how do they look like? How does the final result look like? You may be asked to draw the distributions (but you do not need to provide the equations).
Assumption: Individual measurements are independent given the robot position. Components: 1. Beams reflected off of obstacles (Stop at an obstacle) 2. Beams reflected by persons/caused by crosstalk (Stop short of an obstacle) 3. Random measurements (look like #2) 4. Maximum range measurements (Go through obstacles) Distribution: Goes down like y = -0.5x for a bit, then an inverted parabola , then levels down, and then increases straight up
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 is the impact of Canny's roadmap method in terms of motion planning's complexity
Before Canny's roadmap method, the original complexity for cylindrical cell decomposition was double exponential to the number of dimensions: a^x^d. Canny's approach to cylindrical cell decomposition brings the complexity down to a single exponential a^d. Since we had an algorithm that was single exponential and since we knew that it was as pard as any other problem in p-space, this makes the motion planning problem pspace-complete
What is an Oriented Bounding Box?
Better than axis aligned bounding box, but worse than convex hull (triangles). Stored in 9+6 scalars, slower detection than AABB but faster than convex hull. Similar to bounding capsule with sharp ends.
What nodes and edges of the visibility graph belong to the reduced visibility graph?
Bi-tangent edges, and edges between consecutive vertices on a polygon
What is the issue with applying local search for motion planning
Can end up stuck at a local minimum. Incompleteness. Typically only deform paths in the same homotopic class.
What is the definition of a robot's configuration? What are the degrees of freedom of a robot?
Configuration is the specification of where all points on a robot lie. DoF is the smallest number of real-valued components to describe a configuration
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 main idea behind simple gradient descent and what is its main issue?
Computes the gradient and iterates with a step size until convergence. Problem is that it has slow convergence near minimum point.
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 $P(x|y) = P(x)$.
. How is data association handled in the context of FastSLAM?
Data association is done on a per-particle basis.
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 idea in particle filtering for representing belief distributions?
Distributions are represented as a population of estimates (random samples)
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.
What are the drawbacks of EKF-based approaches for SLAM?
EKF SLAM state vector is much larger than EKF Localization state vector (computationally more expensive), upper bounds on number of landmarks that the map can have due to covariance matrix growing quadratically
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 $A_t\mu_{t-1} + B_tu_t$
Why are POMDPs hard even for relatively small discrete state spaces.
Each belief is a probability distribution, so each value in a POMDP is a function of an entire probability distribution. Even in discrete spaces, the probability distributions are continuous which is a problem because of the huge complexity of belief spaces.
How does FastSLAM maintain representations of the uncertainty regarding the state of the SLAM problem in a landmark-based setup?
Each landmark is represented by a 2x2 Extended Kalman Filter
How does FastSLAM work with grid-based representations? What is the issue that arises in this case? And what is the idea for addressing it?
Each particle represents a possible trajectory of the robot (maintains its own map and updates it upon mapping with known poses). The problem is that each map is big, and therefore the number of particles needs to be kept small. A solution is to compute better proposal distributions by improving the pose estimate before applying the particle filter.
What is the property provided by points along the generalized Voronoi Diagram?
Each point along the generalized Voronoi diagram is as far away from all other obstacle vertices and edges as possible
What types of arcs can arise along the generalized Voronoi Diagram and due to what type of interactions in the polygonal environment?
Edge-edge (straight line), vertex-vertex (straight line), vertex-edge (curved arc)
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 the idea behind policy search? Outline the approach at a high-level.
Evaluate the policies by calculating utilities for some fixed policy until convergence. Improve the policy by using one-step look-ahead with resulting converged utilities as future values. Repeat until policy converges.
Provide an example setup where a potential function may result in a local minimum
Example with a U-shaped obstacle with its mouth open facing left, the start goes into the U and cannot come out even though the goal is behind the U
What is the formal definition of geometric motion planning
Example: Piano Mover's Problem 1) A world W in which either W=R^2 or W=R^3 2) A semi-algebraic obstacle region O in W in the world 3) A semi-algebraic robot is defined in W. It may be a rigid robot A or a collection of m links A_1, A_2, ... A_m 4) The configuration space C determined by specifying the set of all possible transformations that may be applied to the robot. From this, can derive C_{obs} and C_{free} 5) A configuration q_1 within C_free designated as the initial configuration 6) A configuration q_goal within C_free designated as the goal configuration. The initial and goal config together are often called a query pair and designated as (q_1, q_goal) 7) A complete algorithm must compute a continuous path t:[0,1]->C_free such that t(0)=q_1 and t(1) = q_goal or correctly report that such a path does not exist.
Provide the mathematical expression indicating the expected cumulative payoff over a sequence of states $x_1,\dots,x_T$, which have individual rewards $r_q,\dots,r_T$. What is the expected cumulative payoff of a policy?
Expected cumulative payoff: $R_T = E[\sum\limits_{t=1}^T\gamma^tr_{1+t}]$ Expected cumulative payoff of policy: $R_T^\pi(x_t) = E[\sum\limits_{t=1}^T\gamma^tr_{1+t}|u_{t + T} = \pi(z_{t:t + T - 1}u_{1:t + T - 1})]$
What uninformed search technique has the best time complexity? which has the best space complexity? WHat is the time and space complexity?
Fasted uninformed search technique is bidirectional search with O(b^d/2) time complexity. Best spatial complexity is iterative DFS with O(bd) space complexity.
What are the assumptions for the application of the time scaling algorithm in the phase space representation?
For any state s, the robot can maintain its configuration statically and we assume that as s' increases from zero for a given s, there will be at most one switch from admissible to inadmissible
What is the purpose of the Iterated Closest Points (ICP) algorithm and when does it work?
Finds an alignment to determine optimal relative rotation and translation. Requires the starting points to be close enough in order to converge.
What is the premise of a decoupled approach for trajectory planning?
First search for a path in the configuration space, then find a time-optimal time scaling for the path subject to the actuator limit
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 idea in Obstacle-based PRM approach and what issue does it aim to deal with?
Idea that if you sample within a region in C_obs, there may be some useful information. 1) Find a point in C_obs 2) Select a random direction in C_space 3) Find a free point in that direction 4) Find a boundary point between them using binary search It aims to solve the issue of narrow passages with the configuration space
What is the idea of "bounded volume hierarchies" for collision checking?
Ideally, one would check the distance between objects by determining distance between them, seeing if d==0, but it's hard to compute directly. Bounding volume: surround shapes with a sphere-like object for checking. But issue of getting false positives if shape isn't perfectly sphere. From a hierarchy as smaller bounding volumes are added in a tree to encapsulate the object, gain better resolution.
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.
How can we visualize the set of feasible time scaling accelerations at each point of the phase space representation?
It is represented as a cone centered on the tangent vector: the slope of the tangent line in phase space is equal to acceleration
What is the idea behind Monte Carlo Policy Evaluation? What is the update rule at every iteration of the algorithm?
Learns directly from complete episodes of experience, and the value of a state is set to the mean value experiences after this state in experiments. For each state $S_t$ with return $G_t$, $N(S_t) = N(S_t) + 1$ and $V(S_t) = V(S_t) + \frac{1}{N(S_t)}(G_t - V(S_t))$
What is the underlying mathematical principle in Graph-based SLAM? How is it used?
Least squares. Graph-based SLAM uses it to estimate the state which best explains the measurements given.
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 is the relationship between the Levenberg-Marquardt approach and the approaches of gradient descent and Gauss-Newton?
Levenberg-Marquardt regularizes the Gauss-Newton method with a damping factor $\lambda$. When $\lambda$ approaches zero, do the Gauss-Newton method. When $\lambda$ approaches infinity, use gradient descent
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 optimal value of a state in the context of an MDP?
Not described in lecture slides/notes
Describe a high-level methodology for balancing exploitation and exploration in the context of RL.
Not discussed in lecture slides/notes
What are the issues with the beam-based model?
Not smooth for small obstacles and at edges (cause nearby points to have very different likelihoods when sampling) and not very efficient
What is the time complexity of the visibility graph approach?
O(n^2 log(n)) to construct the visibility graph + O(??) to run A*
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 $P(x)$.
Describe a good attractive field for planar navigation
Obstacles are repulsive, goal is attractive
Under which conditions will A* search expand the same nodes as BFS?
On a graph with uniform cost and a heuristic function of h=0
What are the criteria used to compare search algorithms?
Optimality (does it find the optimal solution if one exists), completeness (can it find a solution and can it conclude that no solution exists), time complexity (number of nodes encountered), space complexity (number of unique nodes stored at a given time)
What is the time complexity of trapezoidal decomposition in polygonal environments
Optimality: O(nlog(n))
What is the complexity class of geometric motion planning problem?
PSPACE-complete
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 is the difference between a path and a trajectory?
Path is a feasible route to the goal; trajectory involves scaling velocity along a point to reach the ultimate goal.
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.
Describe the difference between asymptotically optimal RRT* and the original RRT
RRT* connects to the log(n) nearest neighbors instead of just the closest
You may be provided an example of a robot and asked to compute the number of its degrees of freedom or specify the topology of its configuration space
R^n: linear dimension S: circular dimension S^2: torus, etc SE(n): set of all rotations and translations for a single point in n dimensions SO(n): set of all rotations around a single point in n dimensions
Describe how the covariance matrix of an EKF-based approach for SLAM changes over consecutive iterations of the algorithm and why?
The covariance matrix becomes non-sparse over time due to the uncertainty of the features in the map depending on the uncertainty associated with the robot pose.
What is the phase space representation
The phase space is the 2D graph of position vs velocities, where impossible states are represented by obstacles. The upper bound represents the velocity limit curve
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.
How does the bi-directional variant of RRT work?
The same as RRT except it grows RRTs from the start and end nodes, then finds the path when the 2 trees are connected
What does the "Voronoi Bias" property of RRT mean?
The search is biased towards splitting the largest Voronoi region for all the existing vertices in the tree
What is the span(G) of a set of vector fields G?
The set of all linear combinations of vectors in $G$.
