## Abstract

This chapter presents a four-wheel robot’s trajectory tracking model by an extended Kalman filter (EKF) estimator for visual odometry using a divergent trinocular visual sensor. The trinocular sensor is homemade and a specific observer model was developed to measure 3D key-points by combining multi-view cameras. The observer approaches a geometric model and the key-points are used as references for estimating the robot’s displacement. The robot’s displacement is estimated by triangulation of multiple pairs of environmental 3D key-points. The four-wheel drive (4WD) robot’s inverse/direct kinematic control law is combined with the visual observer, the visual odometry model, and the EKF. The robot’s control law is used to produce experimental locomotion statistical variances and is used as a prediction model in the EKF. The proposed dead-reckoning approach models the four asynchronous drives and the four damping suspensions. This chapter presents the deductions of models, formulations and their validation, as well as the experimental results on posture state estimation comparing the four-wheel dead-reckoning model, the visual observer, and the EKF with an external global positioning reference.

### Keywords

- 4WD
- visual odometry
- trinocular sensor
- EKF
- visual observer
- trajectory estimation

## 1. Introduction

Autonomous robots obtain precise information about their surroundings by deploying their sensing devices and developing perceptual tasks to accomplish useful missions. Intelligent robots require to concurrently execute multiple functions such as path planning, collision avoidance, self-localization, tasks scheduling, trajectory control, map building, environment recognition, kinematic/dynamic control, and so forth. Autonomous robots depend on multisensor fusion, which is the process of combining data from the physical sensors into a homogeneous data space.

This chapter presents robot’s visual odometry using sensor data obtained from a homemade radial multi-view device (Figure 1a). For this case, trinocular sensing is divergent; hence, an inherent problem refers to different perspectives in each camera. Besides, the partial overlap between adjacent cameras allows sharing approximately

Parallel trinocular stereo systems had been deployed either to detect the ground [1], or to estimate motion [2]. There are reported works on motion estimation with binocular divergent systems [3], trinocular divergence for visual odometry [4], and divergent visual simultaneous localization and mapping (SLAM) [5]. As a difference from the active sensing modalities for localization [6], and concurrent localization and mapping with parallel multi-view [7], this chapter intends to estimate the posture of a rolling vehicle by exploiting feedback of the rich data fusion that a divergent trinocular sensor provides. Numerous visual odometry algorithms had been reported, using stereo cameras [8], matching multi-frame features [9] and 3D point cloud [10]. Some outdoor visual odometry approaches for urban [11] environments estimate motion tracking by extraction of visual feature points. There are numerous works combining the benefits of visual SLAM algorithms [12, 13, 14] with visual odometry [15], detecting geometrical features [16].

This chapter is organized into to the following sections. Section 2 deduces the sensor fusion observer modeling the trinocular system geometry. Section 3 models the 4WD direct/inverse kinematic solutions. Section 4 deduces the visual odometry formulation and EKF-based control state and estimation. Finally, conclusions are provided in Section 5.

## 2. Trinocular sensing model

This section describes the divergent multi-view geometric model, which basically combines the data of a pair of cameras radially arranged. In addition, this section presents an algebraic analysis of the lateral cameras’ alignment and correction w.r.t. the central camera. The fundamental geometrical relationship of the system divergence was experimentally studied by deploying a homemade prototype onboard a mobile robot, see Figure 2a. Cameras with homogeneous intrinsic parameters are assumed, and cameras are mechanically fixed epipolar. The sensor model’s purpose is to determine the depth information of a point in the scene

To calculate the Cartesian coordinates of

The complementary angles

In the triangle

Thus, for the other cameras’ pair, similar expressions are stated

Hence, the model to express depth information is given by

where,

In addition, the range between camera

Using the depth models

and being *θC* from expression (2) we have

Furthermore, the algebraic deduction along the

thus the following term is stated as

Therefore, the geometry vector model for camera

and the same model is enhanced for camera

Hence, the arbitrary point

**Theorem 1 (Trinocular depth model).** *Let camera* *be the reference for either divergent camera* *or* *. A point coordinates w.r.t. camera* *is* *, and w.r.t. camera* *is* *. Hence, the general depth coordinate model for* *and* *for any divergent pair is*

*for coordinate*

The four points shown by the three cameras may illustrate their transformation, experimentally developed at 1-m distance between the robot and the marks.

## 3. 4WD dead-reckoning controller

Since the visual trinocular approach uses an exteroceptive sensor, we decided to challenge its detection and tracking capabilities with a robot having high holonomic properties. A 4WD robot’s locomotion is prone to experience frequent swift turns resulting in numerous slippages. Thus, a 4WD has to depend more on exteroceptive rather than inner measurements. Comparatively, inner 4WD odometry differs greatly from external visual measurement to infer posture. The proposed dead-reckoning system obtains speed measurements by deploying odometer readings of the four asynchronous drives (Figure 2). A 4WD system is considerably different from a conventional differential dual approach. Moreover, four passive mass-spring-damper suspensions are included in this system (Figure 2b), which varies the inter-wheel distances over time. Particularly, the robot’s 4WD and passive suspensions make the posture observations challenging.

The robot’s dead-reckoning model is fundamental to sense and control position used as feedback, providing motion description as a kinematic reference to match the visual observations when estimating the robot’s motion. The positioning and trajectory control [17], as well as the type of kinematic analysis [18] and the dynamic suspension [19] in this type of robot have been previously reported. The robot’s instantaneous speed

where the wheel’s angular speed

Further, the differential velocity

This model describes that the rotary motion of

The previous equation expresses the conservation of angular motion, and the wheel’s contact point turns w.r.t. the robot’s center,

where for each length

as well as

Thus, substituting

The longitudinal contact point’s distance

where

From Figure 2a, the vertical motion

where the elastic spring restitution coefficient is ^{2}). The damping coefficient is

hence

with integration constant

Substituting the previous expression in (22),

and by algebraically simplifying, the characteristic equation is

and its analytic solution is

As we assume a critically damped system,

Therefore, the damping motion is analytically solved by

where

and

There is a maximal allowable Z-turn displacement speed

In addition, to inversely solve this matrix system, the analytical solution represents the vector of independent control rotary variables

and

The matrix form of the inverse analytical solution for all wheels’ speed under damping variations is stated as

where

## 4. State estimation and feedback position control

This section formulates a deterministic geometric model for visual odometry and the state estimation by an EKF. The proposed model combines pairs of key-points at times

In Figure 3a, the instantaneous angle

and such key-points’ distance

The angle

However, at time

with the key-point’s distance

which is used to obtain the angle

Further, the differential angle

**Proposition 1 (Triangulation odometric displacement).** The robot’s displacement

The triangulation angle

and the orientation angle for each reference

which is required to know the

as well as the

When obtaining numerous key-point pairs simultaneously, the total robot’s displacement is an averaged value of the displacements yielded by all key-point pairs,

Therefore, without loss of generality, for state estimation, let us assume a nonlinear robot’s model state vector

where the state vector is

by developing the dot product from previous expression, we obtain

The measurement model requires the displacements that were inferred through key-point triangulation

where

Thus, the linearized models of the process and measurement are defined next in (54) and (55), such that

and

In addition, the EKF’s prediction models (56) and the correction models (57) are formulated and linearized as

and

Moreover, the recursive Kalman gain for system convergence is

and the state vector of the system is described by

with covariance matrix of the system

Thus, hereafter, the vector and matrix models describing the proposed robot’s system are formulated and incorporated into the conventional EKF. Let us define the robot’s pose vector

Therefore, from the displacement equation (46), which arises from exteroceptive observations, the robot’s Cartesian displacements are

and

as well as

By substituting an averaged Cartesian displacement, one considers

Thus, a measurement is a 3D point arising from either divergent pair Eq. (10) or (11) and deployed by Proposition 1.1. Thus, the robot’s measurement vector model

The process noise covariance matrix

Let us define the nonstationary covariance matrix

the matrix diagonal variances are experimental measurements that describe the trend of the robot motion’s error.

The robot’s motion covariance matrix was obtained experimentally through 500 tests—straight motion, right turns, left turns, clockwise and counterclockwise rotations, with

and

as well as the robot’s yaw statistical measurement model

Furthermore, the measurement noise covariance matrix is

and the matrix

The matrix

Let us summarize the 3D points

### 4.1. State feedback position control

This section describes in six general steps the combined use of the visual observers and the EKF geometric odometer as a recursive feedback for the robot’s positioning control. The robot’s deterministic kinematic model conveys predictions about the robot’s geometry of motion and its observations. Therefore, the deterministic model is used to infer the robot’s motion observations implicitly by the trinocular sensor. The following formulation illustrates how the EKF and the visual odometry model are fed back for the 4WD kinematics.

#### 4.1.1. Kalman gain

The initial estimate of Kalman gain is

#### 4.1.2. Observation

From Proposition 1.1, the visual observers provide

The angle of each key-point

and the local angle of the robot w.r.t. the robot’s previous position is

thus the inferred displacement is

Therefore, the observation vector with Gauss noise

#### 4.1.3. Update estimate

The update estimate is obtained by

#### 4.1.4. Update error covariance

The covariance matrix error dispersion of the system is updated

#### 4.1.5. Deterministic control model

Therefore, the prediction is firstly obtained through the robot’s inverse position control model, from the inverse kinematics equation, Eq. (38)

where, in the previous expression,

This step converges until

#### 4.1.6. State prediction

It follows that state prediction is

and the error dispersion covariance matrix is also predicted at

From the previous step, the estimation process repeats again, going to step one. The previous Kalman process is performed until the robot reaches the goal and the estimation error converges by numerical approximation according to

Therefore, Figure 4a shows the robot’s trajectory obtained by the different comparative approaches conducted in this study. The postures measured by an external visual global reference system are the main references to be compared with. The EKF estimation was obtained by the use of Theorem 1, Proposition 1.1, and Eqs. (71)–(77). In addition, the trinocular key-points used as inputs of the visual odometry model inferred the robot’s displacements, which are shown in same Figure 4a. Furthermore, the dead-reckoning robot system was deployed to infer the robot’s postures and is also shown in Figure 4a. Raw odometry refers to the robot’s dead-reckoning kinematic model used as a mean for direct posture observation through direct kinematics (33) and inverse kinematics (38), but using direct encoder readings by (14).

Figure 4b shows the dead-reckoning and the EKF Cartesian absolute errors, taken as main reference for the visual global reference system. As for the direct dead-reckoning measurements, the absolute error grows exponentially, where the position observation starts diverging before the robot reaches the third turn. As for the EKF model, the Cartesian error w.r.t. the global reference does not diverge but preserves bounded error magnitudes.

As for Figure 4c and d, the EKF’s Cartesian and angular absolute errors w.r.t. the global visual tracker are shown. In Figure 4d, the local minimums and maximums determine the Cartesian regions where the robot performed its turns.

Finally, Figure 5a shows the covariance error behavior obtained at each control loop during the EKF recursive calculations. Figure 5b is a mapping of the measured key-points registered using the state vector (posture) of a robot’s turn to illustrate the map’s divergence.

## 5. Conclusion

This chapter presented a visual odometry scheme for a trinocular divergent visual system that was combined with an EKF for visual odometry estimation. The proposed trinocular geometric model observer geometrically combined adjacent radial views. About

The robot’s trajectory was obtained by different comparative approaches conducted in this study. The postures were measured by an external visual global reference system, which was the main reference system to be compared with. The robotic platform’s kinematics was modeled in terms of a dead-reckoning approach. The direct and the inverse solutions were combined to produce a recursive linearized control model and this was used as the prediction model for EKF estimator. The dead-reckoning robot system was deployed to infer the robot’s postures using directly the four encoders’ readings, with good results obtained only for very short paths. As a comparative perspective, using only the 4WD dead-reckoning system the posture exponentially diverged.

We found bounded Cartesian error for this 4WD robot by deploying the EKF. The trinocular 3D key-points were used as inputs of the visual odometry model that inferred the robot’s displacements by geometrical triangulations.