Kinematic Model Of Tricycle For Mobile Robotics: An Essay.

Present at least the following results in your report:
• Compare the results obtained from the two models and discuss the meaning of the results.
• Show that the robots are moving along the predefined path.
• Calculate the minimum radius of curvature of the circle the vehicle can drive around. Confirm this with simulation results.

Wheeled Robot Assumptions

The tricycle falls under the wheeled robotics in mobile robot locomotion. It is a combination of a variety of hardware and software components. It manages locomotion, sensing, control, reasoning, and communication. The tricycle is able to navigate through its environment while it measures properties of itself and the surrounding. It manages to generate the physical actions and how the robot maps measurements into actions. A tricycle has three wheels; two rear wheels and one front wheel. The steering and power are provided through the front wheel and the control variables are the steering direction and the angular velocity of steering wheel (Dong-Sung, & et al., 2003). The ICC must lie on the line that passes through and it is orthogonal to the fixed rear wheels. When the steering wheel is set to an angle from the straight-line direction, the tricycle will rotate with angular velocity about a point lying a distance R along the line perpendicular to the passing through of the rear wheels. In a synchronous drive robot, each wheel is capable of being driven and steered. The three steered wheels are arranged as vertices of an equilateral. The wheel turn and drive in unison and this leads to a holonomic behavior. All the wheels turn in unison. The vehicle controls the direction in which the wheels point and the rate at which they roll. At such a point, all the wheels remain parallel and the synchronous drive always rotates about the center of the robot (Johann, n.d.).  The synchro drive robot has the ability to control the orientation of their pose directly. In the kinematic modelling, the tricycle falls under configuration space dim C=4. The kinematic model of the wheeled mobile robot provides all the feasible directions for the instantaneous motion. It describes the relation between the velocity input commands and the derivatives of generalized coordinates (a differential model). The mechanically more complex, steered and driven conventional wheel is utilized on Neptune, Hero, and Avatar. These three robots have a tricycle wheel arrangement; the front wheel and it is driven with two rear wheels are at a fixed parallel orientation and are undriven (Nilanjan, & et al., 2004).  

The robots in mobile locomotion communicate with each other or with an outside operator. In this assignment, one of the key attributes of the tricycle is the kinematics. It is a study of the mathematics of motion albeit the consideration of the forces that affect the motion. The kinematics model deals with the geometric relationships that govern the system. It deals with the relationship between control parameters and the behavior of a system (Kanayama, & et al, n.d). 

Results and Observations

It is possible for the tricycle to move forward or back but when it attempts to move to the right or to the left, it tends to slip. No slip occurs in the orthogonal direction of rolling. No translation slip occurs between the wheel and the floor in pure rolling (Vrunda, & et al, 2009). At most, one steering link per wheel with the steering axis perpendicular to the floor. For the differential drive, the posture of the robot,

While the control input is obtained as, 

In a real-world situation, the posture kinematic model is analyzed as,

• Relationship between the control input and the speed of the wheels 

• Kinematic equation
• Non-holonomic constraint

Where H is a unit vector orthogonal to the plane of wheels

The instantaneous center of rotation, 

Where R is the Radius

Straight motion:

There are usually two different types of mechanisms used to maneuver a tricycle. The first type uses the front wheel for both the steering and the drive actions, while the second type uses the front wheel as the steering and the rear wheels as the driving wheels. For this paper, the axial distance, d, is

Some of the wheeled robot assumption are such as the robot is made up only of rigid parts. Each wheel may have a 1 link for steering. The steering axes are assumed to be orthogonal to the ground. The pure rolling of the wheel about its axis is assumed hence no slippage or translational slip and no translation of the wheel. The tricycle is the typical kinematics of AGV.  

1. Derive the kinematic equations for both types of mobile robots

The wheeled mobile robot has been modeled as a planar rigid body that rides on an arbitrary number of wheels in order to develop a relationship between the rigid body motion of the robot and the steering and drive rates of wheels (Nelson, & et.al, n.d). Kinematics differential driven robot and synchronous driven robot,

The forward velocity at the front wheel is simply, v, but due to the kinematic steering, the velocity along the path of the wheel must be, 

This means that the lateral velocity at the front steered wheel must be, 

To find the angular velocity about the CG, which is located at the center of the rear axle as, 

B= baseline between the wheels

V_L – velocity of the left wheel

V_R – velocity of the right wheel

w- velocity forward wheel

R- tricycle turn radius (positive or negative)

The time continuous tricycle model 

For a differential drive, the kinematic model in the robot frame

PART I: Build and Simulate the Kinematic Model of the Robot

Performing an integral on the values, 

We obtain the kinematic equation as,

Simulink to build the kinematic models of the robots and simulate and demonstrate the behavior of the vehicle subject to

Constant driving velocity and a constant steering angle

Constant driving velocity and a linearly changing steering angle

Linearly changing driving velocity and linearly changing steering angle

Plan the driving acceleration and steering velocity profiles for the front wheel steering and back wheel driving robot so that the robot will follow the path.   

The outcome is the same. Consider the x-position where the axle holds together the back wheels and the front wheel. During motion, there is no variation in movement sideways rather the movement is translational in the forward direction. there is no slip in this case. Hence the outcome is the same. The outcome differs in the first section where the steering angle is changed, at that slight moment during the change the automotive experiences a lag in velocity. The change in velocity when moving from forward to reverse is detected.

Part 4: compare the results obtained from the two models and discuss the meaning of the results. Show that the robots are moving along the predefined path. Calculate the minimum radius of the curvature of the circle the vehicle can drive around. Confirm from the simulation results.

Length of the arc 

Total length of travel: 

R=0.8459 m

Using Matlab 

radius =    0.7141 m

Comparing the two simulations, when the tricycle is moving on the reverse it tends to portray a less velocity than when it is moving forward. The main assumption here is that the tricycle is running on a level ground. The simulation results are as shown in the MATLAB simulation file showing there is a lesser velocity for the reverse as compared to the forward movement. The reverse requires a steering angle to be maintained. Change in the steering angle is bound to cause a slower motion as compared to the normal or orthogonal movement where the steering wheel is perpendicular to the ground (Zhang, & et al, n.d). 

The particle filter (PF) algorithm is a Monte Carlo technique for the solution of the state estimation problem. It’s also known as the bootstrap filter or the condensation algorithm where the interacting particle approximations and survival for the fittest. The key idea is to represent the required posterior density function by a set of random samples with associated weights. One can thereafter compute the required posterior density function by a set of random samples and weights determined. 

PART 4: Compare the Results Obtained from the Two Models and Discuss the Meaning of the Results

The use of resampling technique is recommended to avoid the degeneracy of the particles.

Its alternative is the Kalman filter. The complete and demonstration of the working of the particle filter is as demonstrated by the figures below. The figures are able to demonstrate or estimate the positions and velocities of the tricycle as modelled using the particle filter algorithm. The Matlab program to implement the algorithm,

Different locomotion strategies must be adopted depending on the task the robot has to perform and the particular environment the task must be performed in. the tricycle is a terrestrial robot that operates on the ground. A wheeled mobile robot interacts with its environment by moving within the environment in some manner and sensing the environment it moves through. The forward kinematics occur given some control inputs. The inverse kinematics occurs given some desired motion, which control inputs should be chosen in order to obtain the desired motion. The wheeled mobile robots utilize friction and ground contact to enable motion. The wheels rotate about the axle which is considered as the x-axis. The motion is solely in the y-direction hence the measurements of the wheel motion or the odometry, is thereby accurate. For a practical case, the wheel may encounter lateral slip if there is insufficient traction. The rough terrain and bumps, compression and cohesion between the wheel and the ground surfaces often leads to a loss in accuracy. Some of the resultant motion will be in the z direction. there are four landmarks in the space and the laser readings will be given as a vector that contains two columns. The number of rows returned from the sensor depend on how many landmarks the sensor can perceive. In the localization using a Particle Filter, the tricycle is assumed to use its front wheel to steer and back wheel to drive. Some of the assumptions made are:

• The control inputs are corrupted by the Gaussian noises with zeros means and variances
• To estimate the robot position accurately, the external land marks are used to provide measurements that can infer the pose of the robot and assume that a laser range finder is attached to the center of the rear wheel axis.
• The laser sensor measures the distance and the angle between the robot and the landmark. It is assumed that the laser sensor can only measure a distance of two meters and angle of +/- 90 degrees from the heading direction of the car.
• It is assumed that the sampling time of the controller and the sensor is 0.05 seconds.

The non-holonomic constraints limit the possible incremental movements within the configuration space of the robot. The robots with differential drive or synchro-drive move on a circular trajectory and can not move sideways. The holonomic, on the other hand, reduce the control space with respect to the current configuration as well as the configuration space. While analyzing the path constraints it is important to note that the wheels were assumed to rotate without any disturbance, there was enough rotational friction between the wheels and the surface for the rotation of the wheels and that there was no translational slip between the wheel and the surface.  

Conclusion 

In a nutshell, a wheeled mobile robot is able to locomote on a surface solely through the actuation of wheel assemblies mounted on a robot and in contact with the surface. A wheel assembly is a device which provides or allows motion between its mount and surface on which it is intended to have a single point of rolling contact. The movement of the tricycle was analyzed using MATLAB software and the results discussed in the sections above. The movement of a tricycle is only restricted to forward and backward movement as opposed to other multi-directional vehicles that can move even sideways. The change of the steering angle during motion affected the velocity over a given time frame.  

References

Johann Borenstein and Yoram Koren, “Motion Control Analysis Of A Mobile Robot”, Transactions of ASME, Journal of Dynamics, Measurement and Control, Vol. 109, No. 2, pp 73-79.
Dong-Sung Kim, Wook Hyun Kwon, and Hong Sung Park, “Geometric Kinematics and Applications of a Mobile Robot”, International Journal of Control, Automation, and Systems Vol.1, No. 3, pp 376-384, September 2003.

Giovanni Indiveri, “Swedish Wheeled Omnidirectional Mobile Robots: Kinematics Analysis and Control”, IEEE transactions on robotics, VOL. 25, NO. 1, pp 164-171, February 2009.
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Johann Borenstein, “Control and Kinematic Design of MultiDegree-of- Freedom Mobile Robots with Compliant Linkage”, IEEE transactions on robotics and automation, vol. 1, pp 21-35.

Nilanjan Chakraborty and Ashitava Ghosal, “Kinematics of wheeled mobile robots on uneven terrain”, Mechanism and Machine Theory 39, pp 1273–1287, 2004.

S.-F. Wu, J.-S. Mei, and P.-Y. Niu, “Path guidance and control of a guided wheeled mobile robot,” Control Engineering Practice, Vol. 9, Issue 1, pp. 97-105, 2001

Vrunda A. Joshi and Ravi N. Banavar, “Motion analysis of a spherical mobile robot”, Robotica (2009), volume 27, Cambridge University Press, pp. 343–353

1317. Kanayama et al “A Locomotion Control Method for Autonomous Vehicles” IEEE conf. on Robotics and automation:1315-1317.

W.L Nelson& I.J.Cox “Local Path Control for an Autonomous Wheeled Vehicle” IEEE conf. on Robotics and automation:1504-1510.

Zhang Minglu,Pang Shang Xian “Study on the Kinematic Model for a Wheeled Mobile Robot”Intelligence Machine Institute,PR China University.