# Mathematical and computer modeling of rover with elastic suspension longitudinal dynamics

### Аuthors

Krasinsky A. Y.1*, Ilyina A. N.2**, Krasinskaya E. M.3*, Rukavishnikova A. S.4***

1. Moscow State University of Food Production, 11, Volokolamskoe shosse, Moscow, 125080, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
3. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia
4. Moscow Institute of Physics and Technology (National Research University), 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia

*e-mail: krasinsk@mail.ru
**e-mail: happyday@list.ru
***e-mail: nasty.ruka@mail.ru

### Abstract

In the paper we provide a method of constructing an accurate mathematical model and solving a steady motion stabilization problem by the example of a simple rover − a four-wheeled mobile single-link manipulator with elastic suspension. It is an actual problem [16] because Lagrange equations of the second kind are not applicable for such systems. Redundant coordinates and equations of motion in M.F. Shul’gin’s form [1-3] are used to model the mechanical part of the manipulator dynamics. Vector-matrix form of the equations is convenient for analysis and control theory and the theory of critical cases [6-8] applying. The algorithm of solving stabilization problem was developed in [10-11].

The manipulator has to move rectilinearly with a constant speed and keep its clamp (with a camera or a scanner, for example) on a specified height. The system has a geometric constraint as the link is connected with a DC motor gear wheel by an inextensible arm AB. As a control action additional tension at the anchor engine is accepted. The wheels are simulated as two springs.

Solving the problem consists of a few steps:

1. Mathematical modeling using F.M. Shul’gin’s equations.

2. Calculating parameters of a steady motion.

3. The first approximation of the disturbed motion equations around the steady motion is found.

4. The linear substitution [10-11] is used for the system to have a special form of the theory of critical cases [6-8].

5. Manipulator controllability [9] is confirmed.

6. Coefficients of the optimal linear control law are determined uniquely by solving linear-quadratic problem using N.N. Krasovsky’s method [9, 15].

7. System stability is analyzed.

Results of numerical computations showed that the control law provides asymptotic stability with respect to all variables. This algorithm was programmed in MATLAB.

### Keywords:

geometric constraints, redundant coordinates, Shul’gin’s equations, stabilization, steady motion, manipulator

### References

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