Rescue Robot Localization and Trajectory Planning Using ICP and Kalman Filtering Based Approach Ehsan Hashemi1, Maani Ghaffari Jadidi1, Mostafa Yaghobi2 1 Mechanical Engineering Department Mechatronic Research Laboratory (MRL) Qazvin Islamic Azad University, Qazvin, Iran [email protected] , [email protected] 2 Electrical and Computer Engineering Department Mechatronic Research Laboratory (MRL) Qazvin Islamic Azad University, Qazvin, Iran [email protected]

Abstract Fine tuning the robot trajectory, including the robot's target and path in disaster areas, its precision, speed, localization and map generation according to the available data received from higher levels together with decision making on desired speed value of designated robot to achieve these trajectories are main tasks of a rescue robot's control subdivision. Motion control in doubtful environment with the unsuitable localization parameters due to lack of sensor data is a demanding predicament in mobile robots including rescue or omnidirectional robots. Developing kinematics equations besides an appropriate PID controller are essential for an efficient localization system for robots in different field conditions and obstacles. Kalman filtering based approach is widely implemented, both to perform the localization task and to generate speeds. Corrected positioning method by Kalman filter and Extended Kalman filter, EKF, are illustrated and compared with the experimental results of ICP algorithm used in the MRL rescue robot in the presented paper. The calculated errors between the measured real data and the filtered data resulted from KF for straight and curved paths are presented using the proposed dynamic model accordingly.

Keywords: Kalman filter, Extended Kalman filter, Rescue robot, Localization, ICP

Introduction Localization including accurate position and orientation measurements is an essential subject for mobile robots such as Rescue robots. Sensor fusion techniques using global and local sensors with appropriate filtering algorithm are suitable methods for precise self-localization. These techniques are extensively used in developing compatible controllers such as adaptive PID as it is concluded from real-time experimental data [1]. Integrating the measured velocity data from shaft encoders can be used to estimate robot's location and orientation. This method has inevitable increasing errors due to the wheel slippage and the instrument noise, but various calibration methods for mobile

robot odometry system have been executed to reduce the position calculation error [2], [3]. A global reference shall also be used to omit the estimation errors together with the above mentioned techniques to improve the odometry accuracy [4]. Path planning together with the tuning and determination of PID controller parameters, defining appropriate controller parameters in acceleration and deceleration to reach far and near target points and extracting consistent path data including position and orientation are significant matters for autonomous mobile rescue robot in RoboCup competitions. Some noise filtering properties are required in the controller to satisfy any measurement noise associated with the measurement process. Kalman filtering based approach is extensively used, both to generate linear and rotational speeds and to calculate object positions [5]. The Kalman filter is fed with the information from the last state estimations by the filter itself and the sensory information such as odometry results, then the output is filtered and provides relatively reliable location information as well as speed estimates. In addition, future location evaluations are possible for any given time. KF based approaches are recursive which brings the practical specifications that not all data are needed to be set aside in storage and re-processed every time when for example a new measurement arrives. Several efforts have been performed for changing the nonlinear dynamic model of rescue robots models to linear models because of valuable advantages of these models including employing in the Kalman filter based methods, their reliable predictions and simple development. In linearization method, a linear behavior is simulated along several small intervals [6] and nonlinear functions are divided into a specified nominal part and an indeterminate perturbation part. This simulation will result in values which are extrapolated to the general domain. Perturbation Kalman Filter, PKF, is implemented to perform the approximation of the state of nonlinear systems by linearizing its nonlinearities [7]. A nominal trajectory is defined then a first order Taylor series approximation is utilized to linearize the perturbations that occur around the nominal trajectory considering this fact that direction of the derivatives at a point on a surface will affect extrapolation.

acceleration of the wheels correspondingly. The Transform matrix, R in the global coordinate is:

Robot’s dynamic model: Robot dynamic model is presented in the state space with developing relations between inputs, outputs and the state variables using matrix notation. The state space provides an appropriate robot modeling for the EKF method in which robot kinematics and kinetics are also considered. Wheels’ traction forces shown in figure 1 are described as bellow:

Figure 1, Autonomous four wheel drive robot.

w m

⎡1 ⎢0 τ m = k1 ⎢ ⎢0 ⎢ ⎣0

0 1 0 0

⎡0 ⎢0 + k3 ⎢ ⎢1 ⎢ ⎣1

1 1 0 0

φ=

F = [T1 T2 T3 T4 ]

T

(1)

m

F = B tF

In which

(2)

[B] is:

⎡1 1 1 B = ⎢⎢0 0 0 ⎢⎣ l l − l

1⎤ 0 ⎥⎥ − l ⎥⎦

(3)

X which is [xm y m φm ] . The global position matrix T

[xG yG φG ]T

is calculated using the transform matrix.

The acceleration vector in the global coordinate system is mathematically illustrated as the following equation when r is considered as the wheel radius:

( ) rω + R(B ) rω& Where (B ) is Moore-Penrose pseudo g

X&& = mg R& B T *

T * −1

−1

L

g m

T * −1

L

(4)

inverse of the

[ ] ω L is the angular velocity matrix of the wheels [ω L1 , ω L 2 , ω L 3 , ω L 4 ] , and ω& L is the angular

matrix B ,

(5)

0 0 1 1

⎡ . ⎤ 0⎤ ⎢ω m1 ⎥ ⎡0 . ⎢1 0⎥⎥ ⎢ω m 2 ⎥ ⎢ . ⎥ + k2 ⎢ ⎢0 0⎥ ⎢ω ⎥ ⎢ ⎥ m3 1⎦ ⎢ . ⎥ ⎣0 ⎢⎣ω m 4 ⎥⎦ ⎡ . ⎤ 1⎤ ⎢ω m1 ⎥ ⎡1 0 . ⎢0 1 1⎥⎥ ⎢ω m 2 ⎥ ⎢ . ⎥ + k4 ⎢ ⎢0 0 0⎥ ⎢ω ⎥ ⎥ m3 ⎢ 0⎦ ⎢ . ⎥ ⎣0 0 ⎣⎢ω m 4 ⎦⎥

0 0 1 0

⎡ . ⎤ 0⎤ ⎢ω m1 ⎥ . 0⎥⎥ ⎢ω m 2 ⎥ ⎢ . ⎥ 1⎥ ⎢ω ⎥ ⎥ m3 0⎦ ⎢ . ⎥ ⎢⎣ω m 4 ⎥⎦

1 0 0 0

0 0 0 1

0 0 1 0

0⎤ ⎡ω m1 ⎤ 0⎥⎥ ⎢⎢ω m 2 ⎥⎥ 0⎥ ⎢ω m3 ⎥ ⎥ ⎥⎢ 1⎦ ⎣ω m 4 ⎦

(6-1)

(6-2)

.

φ

is the robot rotational velocity about its normal

ωm is angular velocity matrix of the drivers [ω m1 , ω m 2 , ω m 3 , ω m 4 ] . Related K values which mid axis and

couples the drivers’ angular velocity to the torques and describes properties of the robot’s dynamic model to be used in the controller design and implemented in the EKF, are calculated according to the differential equation of the robot’s model in which J L , J m , C L , C m are the wheels’ and drivers’ rotor inertia and damping coefficients. m and n represents the robot mass and the gear ratio.

The position matrix in the local coordinate is introduced as m

Cosφ 0

0⎤ 0⎥⎥ 1⎥⎦

r (ω m 3 − ω m1 ) nl

In which This shows traction forces of wheels 1 and 2 which are located on the same side and also for wheels 3 and 4 which are located on the other side of the robot. The applied forces on the robot are related to the traction force matrix as:

− Sinφ

Using the above mentioned global acceleration vector in equation (4), driver torque matrix, τ m can be expressed as:

.

t

⎡Cosφ R = ⎢⎢ Sinφ ⎢⎣ 0

k1 = J m +

JL r2 + (0.0625m + 1.929 J ) n2 n2

r2 (0.0625m + 1.929 J ) n2 r2 k3 = 2 (0.0625m − 1.929 J ) n C k4 = Cm + 2L n k2 =

(7-1)

(7-2)

(7-3)

(7-4)

The simulated robot’s dynamic model is developed considering effective parameters including driver resistance specifications, torque constants of each driver, drivers’ back EMF, gear ratios and inertias. Inputs are drivers’ torques and outputs are calculated drivers’ angular velocities which used for the next solution of the EKF method as the initial

condition and this iteration continues till the convergence criteria satisfies. The response for the 4-wheel drive model of the robot to the step input is graphically illustrated in figure 2.

covariance are:

K k = Pk− H T (HPk− H T + R )

−1

(

xk = xk− + K k zk − Hxk−

(10)

)

(11)

Pk = (I − K k H )Pk−

(12)

The developed program is used to compare the errors between KF with consideration of process noise and the KF with consideration of both process noise and input noise covariance which is noted by KFU in the figures (3-a) and (3-b).

Extended Kalman Filter Figure 2, Step response of dynamic model of the robot

Kalman Filtering The integration of the wheel revolutions leads to errors in both traveled distance and orientation due to drift and slippage [8]. These errors accumulate over time and result into a large positioning errors. The assumption of KF is that the action and sensor models are subject to Gaussian noise, then the system state and measurements can be described by a linear dynamic system [9]. In practice, this might not always be the case, but it produces a relatively good approximation. [10]. A measurement model is also needed to compare the real data with that which results into a corrected state forecast. Estimated position matrix and error covariance are described as follows: − k

x = Axk −1 + Bu k Pk− = APk −1 AT + Qk −1 + U k −1

(8)

xk− is the preceding state estimate at step k, u k is Input velocity matrix of the robot in the local coordinate system,

A, B are the matrix which relates the states at the

previous time step k-1 to the step k and relates the optional control input

u k to the state x. The former estimated error

Pk− is correlated to the succeeding estimated error covariance at step k-1 as equation (9). Q is the process noise covariance and U is the covariance of the additional input. H is introduced to couple the state to the measurement z k and R is the measurement error covariance

covariance.

matrices of partial derivatives of the f respect to x and input

u k correspondingly. Estimated position matrix and error covariance are described as:

xk− = f ( xk −1 , uk ,0)

(13)

Pk− = Ak Pk −1 AkT + Wk Qk −1WkT + Au , kU k −1 AuT, k

(14)

(9)

In which xk −1 is the subsequent state estimate at the step k-1,

and

Extended Kalman Filter, EKF, is an efficient filtering method due to some disadvantages of using the same nominal trajectory all through the estimation process in PKF, Perturbation Kalman Filter, method which results into a large deviation of nominal state from the real state [11]. This would be attainable with estimating measurement functions at trajectories with the latest state estimates instead of calculation of the Taylor series expansions of the measurement functions at the nominal trajectories. Employed equations for the EKF scheme are described in the following paragraphs in which Ak , Au ,k are Jacobian

Updated Kalman gain, position and error

Wk is Jacobian matrix of partial derivatives of the f respect to input noise w , H k ,Vk are Jacobian matrices of partial derivatives of the H respect to x and measurement noise v . Updated EKF gain, position and error covariance are:

(

K k = Pk− H kT H k Pk− H kT + Vk RkVkT

(

(

xk = xk− + K k zk − h xk− ,0 Pk = (I − K k H k )Pk−

))

)

−1

(15)

(16)

(17)

Execution of EKF on the robot model will result in the following equations. Calculated position matrix is:

⎡ xk− ⎤ ⎡ xk −1 ⎤ ⎡Cosφk −1.T ⎢ −⎥ ⎢ ⎥ ⎢ ⎢ yk ⎥ = ⎢ yk −1 ⎥ + ⎢ Sinφk −1.T − ⎢φk ⎥ ⎢⎣φk −1 ⎥⎦ ⎢⎣ 0 ⎣ ⎦

− Sinφk −1.T Cosφk −1.T 0

⎡. ⎤ 0 ⎤ ⎢ x m ,k −1 ⎥ . ⎥ 0 ⎥.⎢ y m ,k −1 ⎥ ⎢. ⎥ T ⎥⎦ ⎢φ ⎥ m ,k −1 ⎣ ⎦

solutions in which both process noise and input noise covariance are considered. (18)

Estimated error covariance in the EKF after employing robot model is approximated as:

Pk− = AK Pk −1 AkT + Wk Qk −1WkT + U k −1

(19)

In which The Jacobian matrix of partial derivatives of the illustrated f in equation (13) respect to x and input noise w are described in 3×3 matrices as equations (20) and (21).

⎡1 0 − T .xm , k −1Sinφk −1 − T . ym, k −1Cosφk −1 ⎤ Ak = ⎢⎢0 1 T .xm, k −1Cosφk −1 − T . ym , k −1Sinφk −1 ⎥⎥ ⎢⎣0 0 ⎥⎦ 1

(20)

⎡Cosφk −1.T Wk = ⎢⎢ Sinφk −1.T 0 ⎣⎢

(21)

− Sinφk −1.T Cosφk −1.T 0

0⎤ 0 ⎥⎥ T ⎥⎦

The main measurement system implemented to evaluate the filtering task is the corrected laser scanner positioning data from ICP method which update the localization data resulted by KF and EKF. Laser scan matching methods [12], [13] are categorized based on their association: point to point and feature to feature. The point to point matching approach such as ICP, Iterative Closest Point, [14], [15], is to approximate the alignment of two consecutive scans, and then iteratively improve the alignment by defining and minimizing a distance between the scans. Moreover, it does not require the environment to be structured or contain predefined features. The updated position matrix for the rescue robot shown in figure 1 is: − ⎛ ⎡ z1k ⎤ ⎡ xk− ⎤ ⎞ ⎡ xk ⎤ ⎡ xk ⎤ ⎟ ⎜ ⎢ y ⎥ = ⎢ y− ⎥ + K ⎜ ⎢z ⎥ − ⎢ y− ⎥ ⎟ k k k k 2 k ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎜⎜ ⎢ ⎥ ⎢ − ⎥ ⎟⎟ ⎢⎣ z3k ⎥⎦ ⎣φk ⎦ ⎢⎣φk ⎥⎦ ⎢⎣φk− ⎥⎦ ⎠ ⎝ In which K k is:

(

K k = Pk− Pk− + R

)

−1

(22)

(23)

Updated error covariance is attainable using equation (17) which is modified for this model as follow:

Pk = (I − K k )Pk−

(24)

Results and Discussion: Comparing the results for KF, EK, and modified models with the ICP data are presented in this section. Figure 3 shows the differences between errors for KF and KFU

(a)

(b) Figure 3, comparison of errors in the X and Y directions in percent obtained by KF and KFU methods against X locations (a), and Y locations of the robot (b) in meter

As it is shown in figure 3, employing the input error covariance results in to a reasonable decrease in the robot’s location estimation error. X position of the robot deviates from the ICP method data with the value of 2%, then this value changes to 0.35% in the KF method after 50cm robot displacement in the X direction. The error decreases from 1.32% to 0.18% in the KFU. The rate of error reduction in the KFU is higher than the simple KF technique because of utilizing input noise covariance in equations. The same diminishing behavior in the X and Y direction errors are shown in figure 4 for both EKF and EKFU models. Figure 4 shows the comparison between EKF errors with consideration of process noise and the EKF errors with consideration of both process noise and input noise covariance which is noted by EKFU.

Deviation of the calculated results from the predicted path plans in X direction resulted by ICP with the simple KF and EKFU methods could be measured in different zones with different time intervals as it is divided in table 1 and figure 6. The main reason that the errors are very small in some zones and time intervals is that all measurements are relative in the performed maneuvers and there is no absolute measurement technique in the above mentioned calculations.

(a)

(a)

(b) Figure 4, comparison of errors in X and Y directions in percent obtained by EKF and EKFU methods against X locations (a) and Y locations of robot in meter

EKF and EKFU errors from the ICP measurement is less than the KF errors since the robot's dynamic model is used in the filtering analysis and measurement functions at trajectories with the latest state approximations are generated. Robot’s path obtained by a rescue robot movement inside an area is plotted for a go-and-return motion and compared with the ICP results for both KF and EKFU cases as shown in figure 5.

Figure 5, comparison of ICP predicted path by KF and developed EKFU cases

(b) Figure 6, (a): robot’s X position in meter for different time samples (b) Closer view of the curves (a) Table 1: Comparing the maximum KF and EKFU errors in percent for different time intervals and trajectories: 0-18sec Zone I

18-27sec, Zone II

27-48sec, Zone III

48-60sec, Zone IV

X direction; KF method

2.80 %

0.32 %

0.12 %

0.47 %

X direction; EKFU method

0.22 %

0.05 %

0.03 %

0.06 %

Y direction; KF method

3.23 %

4.15 %

0.94 %

0.15 %

Y direction; EKFU method

0.29%

0.28 %

0.14 %

0.03 %

Conclusion:

[6]

As it is obvious from the curves, the calculated errors in comparison with the ICP results increase in the curved paths due to slippage and drift. EKFU method in which covariance of the error matrix is employed provides a good and accurate estimation with minute errors in comparison with the KF. Different errors in percent for various time intervals are evaluated with KF and it is concluded that the increase in the EKFU errors are observed in the curved path or curvature zones of the trajectory. Furthermore, EKFU method makes a superior estimation for the next position of the robot before receiving the new records which improve the approximation speed especially in the map generation tasks. Input error covariance could be estimated off-line as it is done for R and Q matrices which lead to the process speed improvement. Executing the results of the Kalman filtering and EKF optimization presented in this paper together with the robot’s dynamic models to map generation are the future works of the authors.

Acknowledgment This research was supported by the Mechatronics Research Laboratory (MRL) of “Azad University of Qazvin”. Authors gratefully acknowledge the technical support of MRL RoboCup team members.

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

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