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As we did in the case for the robot with front‐wheel steering, we may wish to account for the fact that velocities cannot change instantaneously. Thus, we would introduce as the control variables the velocity rates:
(1.12a)
and
(1.12b)
The system of equations for this kinematic model is now fifth order.
Again we can use the Euler integration method for obtaining a discrete‐time model for this system of nonlinear equations,
(1.13a)
(1.13b)
(1.13c)
(1.13d)
and
(1.13e)
More sophisticated and more accurate methods for obtaining discrete‐time models exist; however, this Euler model may be quite useful if the sampling interval is set sufficiently small. These discrete‐time models may be used for system analysis, controller design, estimator design, and system simulation. More complex models for mobile robots could also include pitch, roll, and vertical motion.
Exercises
1 A front‐wheel steered robot is to turn to the left with a radius of curvature equal to 20 m. The robot is 1 m wide and 2 m long. What should the steering angle be?
2 A differential wheel steered robot is to turn to the left with a radius of curvature equal to 20 m and is to travel at 1 m/s. The width is 1 m and the length is 2 m. What should be the velocities of the right side and the left side?
3 Using the discrete‐time model presented, perform a digital simulation of the front‐wheel steered robot using a steering angle of 45°, a length of 1.5 m, and a speed of 2.778 m/s. Experiment with the sample interval, T and find the maximum allowable value that yields consistent results.
4 Develop a digital simulation for the steered wheel robot modeled in Chapter 1. Assume that the width from wheel to wheel is 1 m and that the length, axle to axle is 2 m. A sequence of speeds and steering angles will be inputs. Include limits in your model so that steering angle will not exceed ±45° regardless of the command. Simulate the robot for straight line motion and for motion when the steering angle is held constant at 45° and then constant at −45°. Simulate several seconds of motion. Use the Euler formula for integration and experiment with the sampling interval. Then use a sampling interval of 0.1 s and see if this sampling interval yields correct results. Plot x vs. t, y vs. t, heading vs. t, and y vs. x.
5 Develop a digital simulation for the differential drive robot, modeled in Chapter 1. Assume that the width from wheel to wheel is 1 m and that the length, axle to axle is 2 m. A sequence of right side speeds and left side speeds will be the inputs. Simulate for straight line motion and for motion when the right side speed is 10% above the average speed (right speed + left speed)/2 and the left side speed is 10% below the average speed. Simulate several seconds of motion. Use the Euler formula for integration and experiment with the sampling interval. Then use a sampling interval of 0.1 s and see if this sampling interval yields correct results. Plot x vs. t, y vs. t, heading vs. t, and y vs. x.
References
1 Canudas de Wit, Carlos, Siciliano, Bruno, and Bastin, Georges (eds.), Theory of Robot Control, Springer, 1996.
2 Corke, P. I. and Ridley, P., “Steering Kinematics for a Center‐Articulated Mobile Robot,” IEEE Transactions on Robotics and Automation, Vol. 17, No. 2 (2001), pp. 215–218.
3 Dudek, Gregory and Jenkin, Michael, Computational Principles of Mobile Robotics, Cambridge University Press, 2000.
4 Fahimi, Farbod, Autonomous Robots: Modeling, Path Planning, and Control, Springer, 2009.
5 Hartley, Tom, Beale, Guy O., and Chicatelli, Stephen P., Digital Simulation of Dynamic Systems, Prentice Hall, 1994.
6 Indiveri, G., “An Introduction to Wheeled Mobile Robot Kinematics and Dynamics,” Robocamp? Padeborn (Germany) (April 8, 2002).
7 Kansal, S., Jakkidi, S., and Cook, G., “The Use of Mobile Robots for Remote Sensing and Object Localization,” Proceedings of IECON 2003 (pp. 279–284, Roanoke, VA, November 2–6, 2003).
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