Modeling of slip for wheeled mobile robots
Summary (2 min read)
I. INTRODUCTION
- Wheeled Mobile Robots (WMRs) are generally modeled as nonholonomic dynamical systems with its wheels assumed to be rolling without slipping.
- Rolling conditions are sometimes violated in tractive maneuvering of WMRs, predominantly due to slipping and scrubbing [3] .
- This paper uses the latter category for simulation studies [9] .
111. ADHESION COEFFICIENT MODEL
- The tractive force, Ft, is generally obtained by the product of the adhesion coefficient and normal force at the point of contact.
- The adhesion coefficient is a function of the wheel dynamics and tractive conditions.
- Unruh [I81 uses a model where the adhesion coefficient is assumed to be constant while Allen and O'Massey [4] assume a model where the adhesion coefficient is a function of the linear velocity.
- These models are useful only when the wheel is locked and the vehicle is skidding.
Adhesion CoefJicient Dependent on Wheel Slip
- This curve has been proposed and verified by Dugoff et al. [8].
- The fall of p,, beyond the peak pop(,,l. results in instability since the tractive force reduces with increasing slip.
- The present adhesion coefficient model based on wheel-slip satisfies this criterion and also enables the representation of combined rolling and slipping in the wheel.
- In view of the above mentioned factors this adhesion coefficient model was used in the simulations.
- Lateral tractive force can also be considered along similar lines [8], [17] .
IV. WMR DYNAMIC MODEL WITH WHEEL SLIP
- The omni-wheel has freely rotating barrels at the periphery and the axis of rotation of the barrels are at an angle to the axis of rotation of the wheel.
- For the planar WMR with three omni-wheels placed at an angular separation of 120", shown in Fig. 3 , one can find the relationship between the wheel variables and the Cartesian variables by using the no slip condition at the three wheels.
- Q are not related through the kinematic relations for 'ideal' rolling shown in (5).
- The authors can make the following observations from the dynamic equations of motion.
- The wheel dynamics alone are given by (1 1).
V. CONTROL OF THE WMR
- The control problem for the WMR may be defined as that of guiding the WMR through a desired Cartesian path with specified terminal points.
- The path tracking performance of the WMR under PID control and a model based control using Cartesian space feedback is discussed.
- The Cartesian space feedback may be obtained either through dead reckoning or referential methods [6] .
- The dead reckoning method uses on-board sensors to estimate the current position of the WMR.
- In the present case, the authors assume that there are on-board sensors such as wheel encoders and accelerometers.
Model-Based Control
- The model based control approach seeks to exploit the model of the system to be controlled to obtain enhanced performance.
- The fundamental idea in this approach is to use the model of the system to be controlled in the control law, such that the resulting error equation is decoupled and linear, and is tunable by PID parameters [7].
- The authors investigate the use of the dynamic equations of motion derived using the condition of 'ideal' rolling, (13) in the modelbased control law.
- One possible rationale for using the 'ideal' rolling model is that it is very hard to model adhesion coefficient and other nonlinearities of the real system.
Error Dynamics
- The error dynamics for the WMR undergoing rolling with slip under the above model based control law is now presented.
- The above condition implies that the ordinary differential equation will be asymptotically stable if the state vector, Y, is of sufficient magnitude ( greater than 11[.
- The errors in WMR path tracking is clearly determined by the quantity 11[-4]-'.
- This is the cause of roll/slip motion at the wheel and hence WMR path deviation.
VI. SIMULATION
- In particular, the authors show that for a single wheel as the adhesion coefficient is increased conditions closer to 'ideal' rolling prevail.
- For the WMR, the performance of PID control is not satisfactory, and the use of 'ideal' rolling model, in model based control, is justified only for high values of adhesion coefficients, i.e., when conditions are closer to 'ideal' rolling.
- The p n versus X curve is approximated by straight lines in the different regions and is given as The differential equations of motion were numerically integrated for obtaining simulation results.
- The equations were found to be "stiff ' [13] , especially for the cases where near rolling conditions were present.
WMR Simulation
- The PID controller and the model based controller were used for simulation.
- Other simulations were also performed with varying Zi,, for a given peak adhesion coefficient value and improvement in performance was observed with increased gain.
VII. CONCLUSION
- Most dynamic models of WMR do not incorporate the effects of wheel slip and traction.
- The equations of motion of the WMR were obtained incorporating the effects of wheel slip.
- The analysis of the error equation suggest that the system is stable under certain conditions.
- Improved performance can be obtained by ensuring that the required tractive forces are available or by increasing controller gains.
- The path deviation of the WMR is small only when the adhesion coefficient is chosen large representing conditions closer to "ideal" rolling.
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Citations
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Cites background or methods from "Modeling of slip for wheeled mobile..."
...The relation between the tractive force and the wheel slip is nonlinear and is a function of the road condition [8]....
[...]
..., u∗(3) was computed in less than 1 minute by solving 4 mpQP problems in the region of the state space X = {x ∈ R| [−8 −8 ] ≤ x ≤ [ 8 ]}....
[...]
...Regarding the second part, several control strategies have been proposed in the literature mainly based on sliding-mode controllers, fuzzy logic and adaptive schemes [13, 95, 11, 143, 144, 109, 8, 142]....
[...]
390 citations
288 citations
Cites background from "Modeling of slip for wheeled mobile..."
...The relation between the tractive force and the wheel slip is nonlinear and is a function of the road condition [2]....
[...]
...Regarding the second part, several control strategies have been proposed in the literature mainly based on sliding-mode controllers, fuzzy logic, and adaptive schemes [2]–[4], [21], [22], [25]–[27]....
[...]
224 citations
180 citations
Cites background from "Modeling of slip for wheeled mobile..."
...To avoid too frequent switchings in θi, in the cases that only the network topology changes, the proposed mechanisms can be adjusted by switching in Pii without changing θi [51], [52]....
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