Kinematic Modelling of a Tricycle Robots Movement
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Added on 2020-05-16
Kinematic Modelling of a Tricycle Robots Movement
Added on 2020-05-16
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Date of submissionStudents name student Id contribution to assignmentUNIVERSITY OR COLLEGEDepartment or faculty
EXECUTIVE SUMMARYThis paper seeks to perform kinematic modelling of a tricycle robot’s movement. Some of themain control factors are the speed of motion of the tricycle as well as the steering angle. Thepaper seeks to derive the kinematic modelling equations in two states where the front wheel doesthe steering and the rear wheel roll over and where the front wheels do both the steering anddriving. The software used for this tests and simulations is the MATLAB R2017a. the MATLABscripts are appended alongside the report for testing purposes. The paper seeks to build andsimulate the kinematic model of the robot. It also seeks to determine the localization using aparticle filter. The discussion section highlights the main assumptions made in regards to thisreport and it analyses the results and observations made in meeting the aims of the tests. Thepaper seeks to determine the trajectory or the path the tricycle takes within a given period of timewhen different parameters are altered to that effect. The derived system model has beencompletely defined to use experimental results obtained for the robot tricycle design. Page 1 of 27
TABLE OF CONTENTSEXECUTIVE SUMMARY.........................................................................................................................1BASIC OVERVIEW...................................................................................................................................3RESULTS & OBSERVATIONS.................................................................................................................6DISCUSSION...........................................................................................................................................21CONCLUSION.........................................................................................................................................24RECOMMENDATIONS AND FUTURE WORK....................................................................................24REFERENCES..........................................................................................................................................25Page 2 of 27
BASIC OVERVIEWThere has been a great development in the artificial intelligence and robotics field. The focusnow is on enabling better control strategies to achieve robust and optimal performance. The threewheeled vehicles are mainly used in the local public transport globally but they tend to lagbehind in terms of speed and steering angles although they have merits in localization andnavigation. In most systems in the industry, the PID is used to ensure control of the system.Some other applications are such as the self-driving car which was designed by google and hasnow been built by other car companies globally [ CITATION Fol13 \l 1033 ]. The car is alreadycommuting in various states and roads nationwide and they are strict in the aerodynamic designfeatures as those could affect the motion. The aerodynamic considerations are basically involvedin the dynamic modelling of a system as opposed to the kinematic modelling as is the focus inthis report[ CITATION Mer16 \l 1033 ]. For instance, currently the Mercedes Benz company isworking on an electric vehicle in relation with a mechanical design. One of the key challengesfaced by many vehicle companies is in the monitoring of the control and stability of a vehicleafter design. The dynamic modelling seeks to discuss the stability factor of the three-wheeledvehicle as designed but the control and speed variations are focused on in the kinematicmodelling. It is crucial to note that the highest number of robot locomotives are based on thedifferential drive model. The model states that the two powered wheels are used to drive therobot as well as change its direction, hence they are used for both driving and steering. Thecontrol strategies for the differential drive model of the robot are completely different and do notapply to other robot designs such as the Ackerman drive robots or the omnidirectional mobilerobots. Another model of locomotion for the three-wheeled robot tricycle is where the steeringand powering is done at the front wheels [ CITATION Miy \l 1033 ]. The advantage of the steeringgeometry has several advantages for the purpose of localization and motion planning. Kinematic steering refers to steering where a robot tries to follow a desired heading by yawing orturning a wheel relative to the vehicle body. The tricycle is considered the single axle vehiclewith a front steering wheel and rolling rear wheels[ CITATION Sin15 \l 1033 ]. A wheeled vehicle issaid to have kinematic steering when a wheel is actually given a steer angle as illustrated in theequations below. A kinematic model for the steering basic vehicle in the inertial frame is givenby the equation, ̇X=vcosψ=Rwωcosψ ̇Y=vsinψ=Rwωsinψψ=vLtanδIn this case, the input control variables are the velocity and the steering angle. These are used inthe experiment and the results are observed as shown in the results section below. The center ofgravity is located in the rear axle for a tricycle. The kinematic equations can be easily simulatedusing the MATLAB software,Page 3 of 27
A tricycle like mobile robot is as shown in the figure below. There are usually two differenttypes of mechanisms used to maneuver a tricycle [ CITATION Zha1 \l 1033 ]. The first type uses thefront wheel for both the steering and driving actions while the second type uses the front wheelas the steering and the back wheels as driving wheels. ωz= ̇ψ=vtLtanδ=vtLThe forward velocity at the front wheel is simply, v, but because of the kinematic steering thevelocity along the path of the wheel must be,vδ=vcosδThe lateral velocity at the front steered wheel must be,vt=vδsinδ=vtanδTo obtain the angular velocity about the center of gravity which is located at the rear axle is,ωz=vLtanδThe time continuous tricycle model using theta is obtained as, ̇x=vcosθ ̇y=vsinθ ̇θ=ωThe time continuous vehicle model while the vs is the velocity of the steering wheel[ CITATIONAlv \l 1033 ].Page 4 of 27
In as much as we are keen on defining the trajectory of the locomotive, there is need todetermine the center of curvature on the path the robot traverse. The curvature of the path is theinverse of the instantaneous radius of curvature. It is centered about the supposed center of acircle. It is known as the ICC and is the center of a circle which passes through the path a givenpoint which has the same tangent and curvature at that point on the path. The absence ofactuators in the wheels gives way to precise localization which in turn helps in better trajectoryfollowing and navigation of the vehicle. The modified mechanical design has its merits in thenavigation, in as much as it poses huge drawbacks in the modeling and control strategies[ CITATION Mur10 \l 1033 ]. The control strategy used in the velocity control is as illustrated insystem block diagram below,The plant is a continuous time system as the data from the set data points and inputs to the plantare discrete in time and use the sampling rate as set in the simulation of the model blocks. Aninstrumental variable system identification approach is used to estimate the control of the system.Trajectories become particularly important in autonomous robotics because the target path to betraversed keeps changing dynamically with time. Hence, the trajectory controller for anautonomous robot has to be more robust and dynamic than that for a manually controlled robot.For the robot design considered in this paper, the trajectory control faces even more challengesbecause of high level planning issues for an autonomous drive. The trajectory control interactswith the steering control. In this work our objective was to design the trajectory control strategywhich feeds the steering control loop[ CITATION Wat \l 1033 ]. The steering control loop has itsown controller whose design is also considered in the paper. For the velocity control system, asmentioned above, we assumed that the robot dynamics are primarily due to the BLDC motorwhich is responsible for the translation. For response to a step input in velocity control, wedesigned a controller based on the model identified. A fast rise time is often the most desirableperformance characteristic for any autonomous mobile robot. Other than the high bandwidth, thecontrol design should be such that the closed loop system is insensitive to external disturbanceswhich arise due to undulations in the road terrain and other environmental disturbances. Toachieve both the design objectives a lead-lag compensator is needed. There are certain distinctadvantages that can be had with a three-wheeled robot design[ CITATION Spr02 \l 1033 ]. The steerusing the front wheel is quite close in working to the design of cars. However, the localizationand navigation of such three wheeled vehicles is completely different. If the drive actuation tothe vehicle is also provided in the front wheel, as is the case for our robot design, the two rearwheels are free. These two wheels can be very effectively used for accurate localization, whichwould have been otherwise impossible in a rear wheel-drive vehicle.Page 5 of 27
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