CSE 478

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Lecture 4, April 1

• Motion models:
  • Capture change in state, \(P(x | a)\)
  • We can use simple models, adding some noise to account for errors
  • Good models incorporate it’s uncertainty
  • Better to overstate uncertainty
• Our MuSHR motion model:
  • Kinematic: assumes “speed”, not acceleration, is the action
  • Assume perfect, flat ground
  • State: vec of x, y, theta (pos, heading)
  • Controls: speed, steering angle
  • New state is old state plus the delta
  • Noise:
    • Add noise to control before propagating through the model
    • We add noise to state after the new prediction
• Euler’s Theorem: TLDR; for any moving, turning object, there is a “center of rotation”, where all of the forces on the car are perpendicular to that center
• Equations:
  • \(\dot{\theta} = w = \frac{v}{L} \tan \delta\)
    • Change in direction
• MuSHR sensor model:
  • \(P(z_t | x_t) \rightarrow P(z_t | x_t, m)\)
    • \(z_t\): observation (laser scan), \(x_t\): state, \(m\): map
  • Sources of stochasticity:
    1. Measurement noise (gaussian around measurement)
    2. Unexpected objects, e.g. map is wrong and someone is in between (add probability between 0 and sensor reading)
       • In class, we’ll use linear probability between zero and obs
    3. Sensor failures (add some noise at \(z_\text{max}\))
    4. Random measurements (add uniform random measurement)

Lecture 5, April 3

• Monte-Carlo methods: random sampling to estimate expectation
• Importance sampling:
  1. Sample from a simple distribution
     • Take samples from proposal dist \(q(x)\)
  2. Reweight the samples to match target distribution \(p(x)\)
     • Reweight samples by \(\frac{p(x)}{q(x)}\)
• Belief updating:
  • Wikihow How-To:
    1. Predict our current beliefs forward with motion model
    2. Reweight our beliefs with observation model
    3. Renormalize our beliefs
  • Resample from our beliefs to prevent getting stuck
• Particle samping issues:
  • Two-room issue:
    • Beliefs will coalesce in a feedback loop
    • E.g. fix: resample less if equal confidence in most weights
    • Fix: low-variance resampling, take initial point and then uniform steps
• Can have variable amounts of particles

Lecture 6, April 8

• Kalman Filtering: belief is a Gaussian, linear Gaussian systems
  • Works with continuous distributions
  • Paramaturized by \(\mu_t\) and \(\sigma_t^2\)
  • Works with linear models with Gaussian noise
  • \(x_t = ax_{t-1} + bu_t + \mathcal{N}(0, \sigma_r^2)\)
  • \(z_t = cx_t + \mathcal{N}(0, \sigma_q^2)\)
• Kalman gain: \(\frac{\text{Var(estimate)}}{\text{Var(estimate)} + Var(measurement)}\)
• In multiple dimensions, we start to use covariance instead of variance
• \(x_t\) is a distribution, but often used in equations as the mean of the distribution
• More efficient than particle filters, but requires linear motion and sensor models

Lecture 7, April 10

• We are not studying the extended Kalman filter
  • Requires us to compute the Jacobian
  • Has issues with highly non-linear functions, only the mean gets the nonlinear update
• Unscented Kalman filter:
  • Easy to get the mean of the new Gaussian - just run the old mean through the non-linear update
  • Getting the variance is harder, we don’t have an easy exact way to do so
  • Instead, we model the distribution (and get it’s variance) via sampling:
    • We get \(2n+1\) “sigma points”, where \(n\) is the dimensionality of the state vector
    • First sigma point is the mean
    • \(x_i = \mu + (\sqrt{(n+\lambda)\Sigma})_i\) for \(i=1,\dots,n\)
    • \(x_i = \mu - (\sqrt{(n+\lambda)\Sigma})_{i-n}\) for \(i=n+1,\dots,2n\)
    • Weight the mean higher, because we know it is true

Lecture 8(?), April 15

• Control: create actuator controls to follow a plan!
• Plans:
  • Tracking a reference trajectory
  • Time-parameterized: timed path
  • Index-parameterized: no time in path
• Open-loop: get commands and run without modification
• Feedback control:
  1. Measure error between reference and current state
  2. Take actions to reduce error
• Choosing reference state for index-parameterized path:
  • Can use nearest point, but can cause issues
  • Pick point with at least \(l\) distance away
• Compute error to reference state:
  • Give reference state it’s own coordinate frame
  • Errors:
    • Along-track error: \(e_\text{at}\) (error in \(x\))
    • Cross-track error: \(e_\text{ct}\) (error in \(y\))
    • Heading error: \(\theta_e\) (error in \(\theta\))
  • See slides for equations
• We will only control steering angle in this course, so we ignore along-track error

Lecture 9, April 17

• Compute control law: \(u = K(x, e)\)
• BANG-BANG CONTROL:
  • Choose between max left and max right
    • (Note: left is positive, right is negative)
  • Think about controllers as a system: this is a pendulum
• PID (Proportional-Integral-Derivative) control:
  • The “benchmark” controller (like linear regression in ML)
  • Very popular in practice
  • \(u = -(K_p e_{ct} + K_i \int e_{ct} dt + K_d \dot{e}_{ct})\)
    • \(K_p e_{ct}\): (Proportional) present term
    • \(K_i \int e_{ct} dt\): (Integral) past term
    • \(K_d \dot{e}_{ct}\): (Derivative) future term
    • The \(K\)’s are the gains for each of the terms
  • Proportional gain:
    • Low: long time to correct, smaller oscillations
    • High: quickly correct, larger oscillations
  • Integral term:
    • Can cap the absolute value of term
    • Discount over time
    • Finite-time
  • Derivative term:
    • Dampens the action by the change in error
    • Reduces oscillations
    • Numeric differentiation is terrible, use analytic derivatives
    • \(\dot{e}_{ct} = V \sin(\theta_e)\)
• How to prove a controller is stable?
  • Stability means error and derivative of error are \(0\) as \(t \rightarrow \infty\)
  • Passive error dynamics: what happens as we don’t apply any controls?
  • With a gradient controller, we’re essentially performing gradient descent on an energy loss landscape
  • This shows any positive controls brings the error to zero
  • There exists a closed-form solution for a provably-correct controller (in theory)

Lecture …, missed

Lecture ?, April 24

• Social navigation: help humans feel comfortable and safe with robots
• Collaboration/teaming: shared autonomy
• Learning from humans: how can humans teach a robot?

April 29

• Global planning: plan high level abstract actions
• Local planning: physical actions being taken for a high-level action
• Configuration space (C-space):
  • Represent state as a point in high-dim space
  • We can use \(\mathcal{S}\) to represent a circle
    • The cross product of two is a torus
• Obstacle specification:
  • Robot is a point in C-space
  • Some of this space is full of obstacles
  • Our obstacle C-space is all points between the intersection of real obstacles and possible robots

May 1

• Explicit graph: precompute all collsions with a graph
• Implicit graph: jointly discover graph and find path within it
• “Best”-first search:
  • Use priority queue with some priority function
  • Completeness: returns a solution if it exists, else returns nothing
• Heuristics: estimate the cost to go
  • Admissibility: heurisic is less than or equal to the goal
  • Consistentency: triangle inequality, \(h(s) \leq c(s, s') + h(s')\)
• Weighted A* multiplies heuristic by \(\epsilon\), solution is within \(1 + \epsilon\) of optimal

May 6

• Create discrete graph approximations of the continuous configuration space
• Graph creation:
  • Sample collision-free configurations as verticies
  • Add neighboring edges between close verticies
  • Use a collision checker to ensure there’s no bad paths
• Collision checking:
  • Use a steer function to find a feasible path
  • Check points along the path for collisions via Van Der Corput sequence
    • Good for catching objects early!
• The boundary value problem relates to connecting configurations - I’m not quite sure
• Sampling strategies:
  • Lattice sampling
  • Uniform random sampling
  • Low-Dispersion sampling (Halton sequence)
• Can interleave graph construction and search

May 8

• DARPA Racer presentation
• Goal: Move as quick as possible with fewest safety inverventions
• Car is a very large version of Mushr
• The autonomy stack is nearly the same as with Mushr
• Perception:
  • Local costmap: map of obstacle costs
• Need to convert local perspective to birds-eye view for planner
  • Pinhole -> voxel -> meanpool
  • Use a backbone to complete the full map
  • Use image inpaiting for prediction
    • Note: this is different from segmentation, does inference on outside data
  • Costmap is weighed average of features
• Planning:
  • A* search (working on D*…what is that?)
• MPC:
  • Have a set of possible trajectories
• Sensing uncertainty:
  • Aleatoric: uncertainty due to environment (e.g. occlusion)
  • Epistemic: lack of support from training data (ML uncertainty)
  • Where these pop up:
    • Fluctuating obstacles
    • Consistent inaccurate predictions
    • Myopic perception
  • Determinization: sample or threshold stochastic world into a deterministic one
• Super fucking cool

May 20

• RL Lecture: typical MDP stuff
• Inverse RL: learn reward function from (optimal policy) examples
• (Rough) algo:
  • Start with some reward function \(r\)
  • Learn policy for \(r\)
  • Compare policy to demonstrations
  • Update \(r\)
  • Repeat
• Linear reward function: reward is a linear function of the features
• Linear inverse RL: update \(r\) via difference in feature expectations