Contextualización del pueblo indígena muisca

Experimental validation was done with a handheld IMU moved around in a VI- CON motion capture room. We moved a Microstrain GX3-25 IMU with VICON markers around for about one minute, in a trajectory snaking from one corner to the opposing corner of the room. Motion capture provides a reference measure- ment of the sensor package’s position in some world frame. The purpose of these

Update frequency [Hz] 100 101 102 103 104 105 0.016 0.004 0.001 Legend 10-2 10-1 100 101 102 mean( max(δb / b+ϵ) )

Figure 4-4: Accuracy of inferred inertial sensor biases using compensation gradi- ents computed at higher and higher speeds. Three lines show 1σ level for ran- domly generated gyroscope, rad/s, and accelerometer, m/s2_{, biases over a single}

interpose period. The y-axis shows the mean of the maximum among gyroscope and accelerometer biases divided by the actual bias, for 500 Monte Carlo runs. Bias error is denoted true sensor bias minus inferred bias, δb = b − ˆb. Note for this plot, a stabilizing term  = 1e − 5 has been added to avoid division by zero.

validation tests are not to demonstrate an independent robot localization solution, but to investigate retroactive bias estimation.

Once confidence has been gained in our ability to estimate biases, we present localization and mapping solutions using inertial odometry in combination with aiding measurements such as a camera or kinematics. Results from a localization solution is presented in section 8.2.2.

For analysis, we use ground truth position, but not orientation, measurements (3 degrees of freedom) as priors to every 4 seconds (fourth pose) in the trajectory, as shown in Fig. 4-5; this setup is similar to EKF style INS/GPS systems, and used here to validate and demonstrate the bias estimation characteristics of the pro- posed inertial odometry factors. Incremental solutions to the constructed SLAM problem were computed with a modified iSAM1.7 algorithm [116] using Powell’s Dog Leg trust region method extension [197, 198].

Figure 4-5: Sample factor graph used for validation of preintegral inertial sen- sor compensation model. Green poses are node points separated by pure inertial odometry constraints, red unary factors represent 3DOF position constraints from a GPS or Vicon position. 0 20 40 60 −0.004 −0.002 0 0.002 0.004 Time [sec]

Gyro bias [rad/s]

b

ω

+ε − ε

Figure 4-6: Gyro bias perturbation test to validate the Taylor expansion manifold assumption.

Retroactive bias estimation

The continuous time inertial odometry residual model relies on a local Taylor ex- pansion, eq. (4.15). To experimentally explore that this assumption holds, we pro- pose measuring a sufficient region of interest around the operating point of the expansion. We note the Taylor expansion represents an approximation of the man- ifold on which bias and gravity terms influence the inertial preintegral terms. Our region of interest is defined by the expected inertial sensor bias performance and availability of aiding information.

Using the Microstrain data in the VICON room data, as shown in Fig. 4-5, we quantify the repeatability of gyroscope bias estimation when it is perturbed by a known amount . The same SLAM solution is computed for three different cases while we artificially inject or remove sensor biases. The first is the nominal case, where all variables are estimated and bias estimates supposedly represent native errors in the system. We use this as a control experiment.

offset of  = ±0.0025 rad/s added to the x − axis rate measurement. The three traces in Fig. 4-6 correspond to the resulting gyroscope bias estimates for each test. Visually we can see the distinct pattern repeated with the expected ±0.0025 rad/s offset. More quantitatively, if we look at the ratio of the remaining error:

std. dev.    bx− ˆbx  −  bx  ≈ 15%. (4.85)

The error associated with the 15% discrepancy is attributed to the stationary bias assumption; first order bias only compensation, eq. (4.15); ignoring of accel- eration biases in eq. (4.51); and errors in the computation of matrix exponentials. Also note the 15% average includes possible division by zero, when the control bias estimate is zero. We therefore state the repeatability is likely much better than 85% for gyro terms, and note the gyro bias coupling is the most complicated.

Paying more attention to the sensor calibration process while estimating all sen- sor bias terms, we can gain insight into the overall operation of inertial odometry residual inference. Fig. 4-7 shows the accelerometer and gyroscope bias estimates for the same Microstrain IMU trajectory, as enforced by XYZ position measure- ments from a VICON system once every 4 seconds. We instantiate a new pose ev- ery 1 second. The x-axes show the pose number for three difference time instances in the IMU trajectory of around 1 minute, namely at 27 s, 42 s, 53 s. We specifically note the dashed and solid lines are sensor bias estimates for the same trajectory but at difference times. The curves show the bias estimates updating retroactively as observability changes. We also note, especially with gyro biases, that the ear- lier poses are better constrained with less variation as the trajectory evolves. The z gyro bias (horizontal) sees a significant update (change in observability) after 42.5 s, visible as the jump in the solid to dashed blue curve.

We emphasize that no VICON orientation measurements where used to esti- mate world frame rotations, yet we are able to recover a gyro bias estimates – which were baked into the preintegrals – with strong repeatability. Cross coupling of gravity into horizontal velocity states allow us to observe pitch and roll, while sev- eral separate reference position measurements constrain heading with respect to the world frame.

The retroactive bias estimates shown in Fig. 4-7 are interesting from an observ- ability perspective. We note in Fig. 4-7 the estimates still vary between the 27 s and 53.5 s systems. In contrast to Fig. 4-7, where the x gyro bias estimates are pretty stable between the 42.5 s and 53.5 s systems. Note a larger change in the

0 10 20 30 40 50 60 0.02

0.04 0.06

Incremental batch accel bias [m/s2]

x 0 10 20 30 40 50 60 0 0.05 y 0 10 20 30 40 50 60 −0.02 0 0.02 z Time [s] t=27s t=53.5s 0 10 20 30 40 50 60 −0.002 0 0.002 x

Incremental batch gyro bias [rad/s]

0 10 20 30 40 50 60 −0.002 0 0.002 y 0 10 20 30 40 50 60 0.009 0.01 0.011 z Time [s] t=42.5s t=53.5s

Figure 4-7: Left, inferred accelerometer bias estimates which are improved as more information is collected in the factor graph. The solid lines represent the incremen- tal SLAM solution at the 27 s point, the dashed line is the same trajectory but at 53.5 s into the trajectory. Incremental solutions were computed with a modified version of iSAM1.7. Right, smoothed gyroscope bias estimates as more informa- tion is collected along with growth in the trajectory history.

z gyro bias. The stability of the bias estimates come down to observability of the different error characteristics. The x adn y gyro biases are well observed at 42.5 s. Improved bias observability, more sensor data or future loop closures all allow for better inertial bias estimation.