Active SLAM for a planar robot
Ziqi Lu and Tao Pang
 Hold right mouse button: pan.
 Mouse wheel: zoom.
Structure
FrontEnd

A 2D robot with two translational DOFs: \(X_B^W = [x, y]\)

Noisy measurements of landmarks’ distance \(d\) and bearing \(b\).
BackEnd

Landmarkbased SLAM solved with LevenbergMarquardt.
Active SLAM with horizon \(L\)
Computes robot control \[u^*_{k:k+L1} = \text{argmin} J_k(u_{k:k+L1}),\]
where \(J_k(u_{k:k+L1})\) accounts for
 control effort \(\sum_{i=1}^L \u_i\^2\),

robot pose uncertainty (sum of covariance trace),

Distance to current goal \(\X^W_{B_{k + L}}  X^W_{B_{\text{goal}}}\^2\) + ...
: robot goals.
: landmark ground truths.
: landmark beliefs.
: current robot position (ground truth).
: robot sensing range.
Landmark measurements.
:Robot pose beliefs. \(k\): current time step.
:landmark beliefs.
Additional details:
 Robot dynamics: \( X^W_{B_{i+1}} = X^W_{B_{i}} + u_i \).
 Measurement noises are Gaussian for \(d\) or wrapped Gaussian for \(b\).
: information matrix of current robot pose and landmark beliefs.
Dependencies
Control effort
Pose uncertainty
Distance to goal
Using existing libraries are nice, but...
 iSAM/GTSAM minimizes cost functions of the form \(\r(x)\^2\). It seems difficult to massage (41) into a nonlinear least square...
 ROS involves multiple processes communicating with each other though message passing, which makes debugging harder and results nondeterministic.
 As a result, we implemented Active SLAM planner using mostly numpy, with pydrake and pytorch for gradient computation.
 As (41) shares the same Jacobian computation as SLAM Backend, we also implemented SLAM backend as a sanity check.
 Our frontend is simple enough to be implemented with numpy.
We need \(u^*_{k:k+L1} = \text{argmin} J_k(u_{k:k+L1})\), where
Backend (landmarkbased SLAM)
: robot goals.
: landmark ground truths.
: landmark beliefs.
: current robot position (ground truth).
: robot sensing range.
: robot covariance ellipse.
Backend (landmarkbased SLAM): Loop closure
: robot goals.
: landmark ground truths.
: landmark beliefs.
: current robot position (ground truth).
: robot sensing range.
: robot covariance ellipse.
Oasis: map for Active SLAM
 The environment is devoid of landmarks except near the origin.
 Following the given path naively leads to large variance in the robot's belief of its position.
: robot goals.
: landmark ground truths.
: landmark beliefs.
: current robot position (ground truth).
: robot sensing range.
: robot trajectory belief.
: robot trajectory ground truth.
Small \(\alpha\): reaching goal.
Large \(\alpha\): Reducing uncertainty.
Navigating oasis with Active SLAM
When \(\alpha\) is small, the robot mostly follows the goals.
Small \(\alpha\): reaching goal.
Large \(\alpha\): reducing uncertainty.
time
Navigating oasis with Active SLAM
As alpha grows, the robot is driven away from goals and towards landmarks.
Small \(\alpha\): reaching goal.
Large \(\alpha\): reducing uncertainty.
time
A loop closure is achieved, reducing the variance of the entire trajectory.
Navigating oasis with Active SLAM
The robot reaches its goal with much smaller variance.
Small \(\alpha\): reaching goal.
Large \(\alpha\): reducing uncertainty.
vs.
Without active SLAM.
Computing the gradient of \(J_k(u_{k:k+L1})\)
Control effort
Pose uncertainty
Distance to goal
 Optimizing \(J_k(u_{k:k+L1})\) is done by gradient descent, therefore we need to compute the gradient quickly.
 We experimented with three ways to compute the gradient of \(J_k(u_{k:k+L1})\):
 Numerical differentiation.
 Forwardmode automatic differentiation, using Eigen::AutoDiffXd (python bindings provided by pydrake).
 Reversemode automatic differentiation, aka backdrop, using pytorch.
Computing \(\nabla_x f(x)\) for \(f(x) : \mathbb{R}^n \rightarrow \mathbb{R}\)
Our objective function \(J_k(u_{k:k+L1})\) shares the same form as \(f(x)\).
Numerical differentiation
import numpy as np
def calc_derivative_numerical(x, f):
# derivative of f
df = np.zeros_like(x)
n = len(x)
h = 1e3
for i in range(n):
x_plus = x.copy()
x_minus = x.copy()
x_plus[i] += h
x_minus[i] = h
df[i] = (f(x_plus)  f(x_minus)) / 2 / h
return df
import numpy as np
from pydrake.math import AutoDiffXd as AD
def calc_derivative_forward(x, f):
n = len(x)
# initialize array of x with scalar type AD.
x_ad = np.zeros(n, dtype=object)
for i in range(n):
dx_i = np.zeros(n)
dx_i[i] = 1.
x_ad[i] = AD(x[i], dx_i)
# evaluate f with x_ad.
y_ad = f(x_ad)
return y_ad.derivatives()
Reversemode AutoDiff
import torch
def calc_derivative_reverse(x, f):
# initialize array of x as tensor.
x_torch = torch.tensor(x, requires_grad=True)
# evaluate f with x_torch.
y_torch = f(x_torch)
# backprop.
y_torch.backward()
return x_torch.grad
Forwardmode AutoDiff
x0 = AD(3, [1, 0])
x1 = AD(4, [0, 1])
y = x0 + x1 # AD(7, [1, 1])
value \(\in \mathbb{R}\)
partial derivatives\(\in \mathbb{R}^n\)
Computes \(f(\cdot)\) \(2n\) times.
Computes \(f(\cdot)\) \(n + 1\) times.
Computes \(f(\cdot)\) \(2\) times.
Gradient of \(J_k(u_{k:k+L1})\): computation time
\(u_{k:k+L1} \in \mathbb{R}^{20}\) (\(L=10\)), \(k=1\), 9 landmarks.
 Numerical: 502 ms ± 2.23 ms
 Forward: 35.4 s ± 598 ms (!)
 Backward: 423 ms ± 2.61 ms
Control effort
Pose uncertainty
Distance to goal
 Computation cost grows with the current time \(k\) and the number of observed landmarks.
Profiling the computation of \(J_k(u_{k:k+L1})\)'s gradient: Numerical
Control effort
Pose uncertainty
Distance to goal
Most of the time is spent on computing \(f(\cdot)\), as expected.
Profiling the computation of \(J_k(u_{k:k+L1})\)'s gradient: Forward AutoDiff
Control effort
Pose uncertainty
Distance to goal
 Most of the time is spent on matrix multiplication...
 numpy may not have an efficient backend to deal with vectorized operations on AutoDiffXd.
Profiling the computation of \(J_k(u_{k:k+L1})\)'s gradient: Reverse AutoDiff
Control effort
Pose uncertainty
Distance to goal
Evaluating \(J(u)\) and computing \(\nabla_u J(u)\) takes the same amount of time, as expected.
Computing the gradient of \(J_k(u_{k:k+L1})\): more complex cases
\(u_{k:k+L1} \in \mathbb{R}^{20}\) (\(L=10\)), \(k=9\), 28 landmarks.
 Numerical: 1.86 s ± 3.52 ms
 Forward: N.A.
 Backward: 1.42 s ± 3.46 ms
\(u_{k:k+L1} \in \mathbb{R}^{20}\) (\(L=10\)), \(k=46\), 15 landmarks.
 Numerical: 1.69 s ± 31.1 ms
 Forward: N.A.
 Backward: 754 ms ± 2.35 ms
Making it faster:
 Sparsity
 GPU
Thanks!
 Hold right mouse button: pan.
 Mouse wheel: zoom.
Structure
FrontEnd

A 2D robot with two translational DOFs: \(X_B^W = [x, y]\)

Noisy measurements of landmarks’ distance \(d\) and bearing \(b\).
BackEnd
 Landmarkbased SLAM solved with LevenbergMarquardt.
Active SLAM with horizon \(L\)
Computes robot control \[u^*_{k:k+L1} = \text{argmin} J_k(u_{k:k+L1}),\]
where \(J_k(u_{k:k+L1})\) accounts for
 control effort,

robot pose uncertainty,

Distance to current goal.
Landmark measurements.
:Robot pose beliefs.
:landmark beliefs.
Additional details:
 Robot dynamics: \( X^W_{B_{i+1}} = X^W_{B_{i}} + u_i \).
 Measurement noises are Gaussian for \(d\) or wrapped Gaussian for \(b\).
:landmark beliefs.
: information matrix of current beliefs.
:landmark beliefs.
Backend (landmarkbased SLAM): Loop closure
: robot goals.
: landmark ground truths.
: landmark beliefs.
: current robot position (ground truth).
: robot sensing range.
: robot covariance ellipse.
Oasis: map for Active SLAM
: robot trajectory belief.
: robot trajectory ground truth.
(a)
(b)
Navigating oasis with Active SLAM
As alpha grows, the robot is driven away from goals and towards landmarks.
Small \(\alpha\): reaching goal.
Large \(\alpha\): reducing uncertainty.
time
A loop closure is achieved, reducing the variance of the entire trajectory.
\(\alpha = 0\)
\(\alpha=1\).
(a)
(b)
(b)
(c)
16.485 presentation
By Pang
16.485 presentation
 236