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# Passivity-based control of an omnidirectional mobile robot

- Chao Ren
^{1}, - Yi Sun
^{2}and - Shugen Ma
^{1, 2}Email author

**Received:**24 January 2016**Accepted:**18 April 2016**Published:**7 July 2016

## Abstract

This paper studies passivity-based trajectory tracking control of an omnidirectional mobile robot. The proposed control design is simple to be implemented in practice, because of an effective exploitation of the structure of robot dynamics. First, the passivity property of the prototype robot is analyzed. Then the control system is designed based on the energy shaping plus damping approach. We find that the prototype robot itself has enough damping forces. As a result, only energy shaping is needed in our proposed controller, while the damping injection is unnecessary for our robot. In other words, the disadvantages of differential feedback, such as amplifying the measurement noise, can be avoided. Globally asymptotic stability is guaranteed. Both simulations and experimental results show the effectiveness of the proposed control design.

## Keywords

- Passivity
- Dynamic control
- Omnidirectional mobile robot

## Background

Omnidirectional mobile robots (OMRs) are becoming increasingly popular in many applications. OMRs have the ability to move simultaneously and independently in translational and rotational motion. Therefore, they are especially useful in environments congested with static and dynamic obstacles and narrow aisles, such as hospitals, warehouses, residential homes, and sheltered workshops for disabled people.

In the literature, many studies have been conducted in the dynamic model-based control design for OMRs. In [1], a feedback linearization approach, i.e., resolved acceleration control, was applied to an OMR with lateral orthogonal-wheels. In [2], a linear optimal tracking controller was designed, in which the main idea is to simplify the dynamics of the three-wheeled OMR as a linear time invariant model by using the kinematics. In [3], based on a dynamic model without considering motor dynamics, an adaptive motion controller was synthesized via the adaptive backstepping approach. In [4], feedback linearization strategy was used to compensate the static friction, and then a model-predictive control scheme was applied to trajectory tracking control of a three-wheeled OMR. In [5], generalized proportional integral (GPI) observer was employed to design the controller, in which the unmodeled dynamics and nonlinearities, etc., are considered as a perturbation input. In [6], a smooth switching adaptive robust controller was proposed to switch between a nominal adaptive linearizing controller and a deputy adaptive sliding-mode controller. However, these methods actually stem from the well-known control theory, thereby neglecting the natural structure imposed by the physical character of the robot system. One common problem of these methods is that the differential feedback is necessary.

On the other hand, passivity is one of the most fundamental properties of robotic systems [7]. It has been a very powerful concept in many control problems in robotics: stability analysis [8, 9], teleoperation control [10–12], flexible robot control [13–15], to name a few. However, so far, it has been overlooked for the control problem of OMRs.

In this paper, a passivity-based trajectory tracking control system is designed for a three-wheeled OMR with MY wheel-II. The proposed control design is simple to be implemented in practice, because of an effective exploitation of the structure of robot dynamics. The passivity property of the open-loop dynamic system is analyzed based on a dynamic model. We find that the robot is a fully damped system and the damping forces of the robot itself are large enough due to the large gear reduction ratio of motors. Then energy shaping plus damping approach is applied to our robot, wherein only energy shaping is necessary due to enough damping forces of the robot itself. In other words, the disadvantages of differential feedback, such as amplifying the measurement noise, can be avoided. Globally asymptotic stability is guaranteed. Both simulations and experimental results show the effectiveness of the proposed control design.

## Methods

In this section, we first derive a dynamic model for the omnidirectional mobile robot, and then the passivity property of the open-loop robot dynamic system is analyzed.

Nomenclature

World coordinate frame | |

\({\mathbf {q}} = \left[ {\begin{array}{lll} x &\quad y &\quad \theta \\ \end{array} } \right] ^{\mathrm{T}}\) | Robot position and orientation angle |

\({\mathbf{V}}_{\mathrm{M}} = \left[ {\begin{array}{lll} {V_x } &\quad {V_y } &\quad {\dot{\theta } } \\ \end{array}} \right] ^{\mathrm{T}}\) | Robot translational velocity and rotational angular rate |

Mechanical constants | |

| Robot mass |

\(I_\mathrm{v}\) | Robot moment of inertia around the mass center of the robot |

\(I_\mathrm{w}\) | Wheel moment of inertia around the wheel shaft |

| Wheel radius |

\(D_{\mathrm{in}}\) | Inner contact radius |

\(D_{\mathrm{out}}\) | Outer contact radius |

\(L_{0}\) | Average contact radius |

\(I_0\) | Combined moment of inertia of motor, gear train and wheel referred to the motor shaft |

\(b_0\) | Combined viscous friction coefficient of the motor, gear and wheel shaft. |

\(k_{\mathrm{b}}\) | Motor back EMF constant |

\(k_{\mathrm{t}}\) | Motor torque constant |

\(R_{\mathrm{a}}\) | Motor armature resistance |

| Gear reduction ratio |

Note that, as shown in Figs. 1 and 2, each MY wheel-II assembly has two contact points with the ground, and therefore two contact radiuses exist for each wheel (i.e., \(D_{\mathrm{in}}\) and \(D_{\mathrm{out}}\)) [17]. In our previous work [17], a continuous dynamic model including the DC motor dynamics has been derived for the robot by using an average contact radius (i.e., \({L_0} = {({D_{\mathrm{in}}} + {D_{\mathrm{out}}})}{/}2)\), while the resulting parameter errors are considered as perturbations to the nominal dynamic model. In addition, it is assumed that no slippage is between the wheel and the motion surface. The coulomb and viscous friction, dead-zone and backlash are also unmodeled.

Note that \({\mathbf {D}}\) is symmetric and positive definite and thus \({{\dot{\mathbf{q}}}^{\mathrm{T}}}{\mathbf {D}}{\dot{\mathbf{q}}} > 0\). Therefore, according to the standard passivity definition [19], (5) defines an output strictly passive mapping from the virtual control input \({\varvec{\tau }}\) to \({\dot{\mathbf{q}}}\). Note that, the passive mapping from the real control input \({\mathbf {u}}\) to \({\dot{\mathbf{q}}}\) cannot be guaranteed.

There are two steps in the passivity-based control approach, i.e., energy shaping and damping injection. The first step is an energy shaping stage where the potential energy of the system is modified in such a way that the new potential energy function has a global and unique minimum in the desired equilibriums. Second, a damping injection stage where the dissipation function is modified to ensure global asymptotic stability. For Eq. (5), it is observed that the potential energy is absent. The energy shaping is thus indispensable. However, the damping injection stage can be avoided if the dissipative forces of the robot itself \({\mathbf {D}}{\dot{\mathbf{q}}}\) are large enough to satisfy the control system performance requirements. In other words, by making use of the structure of the robot dynamics, the controller design can become easy and simple. Indeed, it is shown in our simulation and experiments that the controller is able to achieve good performance even though no damping is injected into the system.

###
*Remark 1*

It can be seen that the dissipative force \({\mathbf {D}}{\dot{\mathbf{q}}}\) is related with the gear reduction ratio and wheel radius. More specifically, the dissipative force has a positive correlation with the gear reduction ratio *n* and an inverse correlation with the wheel radius *r*.

###
*Remark 2*

It is worth pointing out that the robot is a continuous linear dynamic system when the robot moves only with translational motion, and no parameter uncertainties of the contact radius \(L_0\) exist in the robot dynamics. However, the parameter uncertainties in the robot contact radius \(L_0\) will appear in the robot dynamics if the robot moves with rotational motion. In fact, the robot is an autonomous switched nonlinear system in this case [17].

## Control system design

### Control design

In this section, we derive a trajectory tracking controller only with the energy shaping. The tracking control problem is formulated as follows: Given a reference trajectory \({{\mathbf{q}}_d}(t) = {\left[ {\begin{array}{lll}{{x_d}}&\quad {{y_d}}&\quad {{\theta _d}}\end{array}} \right] ^{\mathrm{T}}}\), which is bounded and twice continuously differentiable, find a control input \({\mathbf{u}}(t)\) such that the responses of the robot, \({\mathbf{q}}(t) = {\left[ {\begin{array}{lll}x&\quad y&\quad \theta \end{array}} \right] ^{\mathrm{T}}}\), converges to \({{\mathbf{q}}_d}(t) = {\left[ {\begin{array}{lll}{{x_d}}&\quad {{y_d}}&\quad {{\theta _d}}\end{array}} \right] ^{\mathrm{T}}}\) for any initial condition.

It is known that differential feedback usually introduces the problem of noise amplification. Therefore, one advantage of the proposed controller is no differential feedback. It is also noted that only the measurement of the rotational velocity (\(\dot{\theta }\)) is needed in the controller, while the robot translational velocity (\(\dot{x}\) and \(\dot{y}\)) is not used [see (5)]. This is another advantage of the proposed controller. For example, in the well-known computed torque control, the measurement of the robot velocity \({\dot{\mathbf{q}}}\) is indispensable.

In addition, it can be seen that the closed-loop error dynamics (8) does not result in decoupled linear systems. The damping forces of the robot itself are also reserved. These are the main differences from feedback linearization approaches, such as the well-known computed torque control.

###
*Remark 3*

Although there are parameter uncertainties in \({L_0}\) since the real contact radius of each wheel is \({D_{\mathrm{in}}}\) or \({D_{\mathrm{out}}}\), the parameter uncertainties are not considered in the controller design, in order to facilitate the theoretical analysis. It is shown in our experiments that the control system performs well even though the parameter uncertainties appear when the robot moves with rotation.

###
*Remark 4*

### Stability analysis

## Simulations

In this section, simulations of the proposed control system are implemented in MATLAB/Simulink. The robot physical parameters used in the proposed control system design are as follows: \({m} = 33\) kg, \(I_{{\rm v}} = 1.35\) \({\mathrm{kg}}\,{{\mathrm{m}}^2},\,R = 0.06\) m, \(D_{\mathrm{in}} = 0.147\) m, \(D_{\mathrm{out}} = 0.236\) m, \(I_0 = 3.15\times 10^{-5}\,{\mathrm{kg}}\,{\mathrm{m}^2},\,k_{\mathrm{t}} = 0.0292\, {\mathrm{N}}\,{\mathrm{{m}/A}},\,k_{\mathrm{b}} = 328\) rpm/V, \(n = 185.7,\,b_0 = 1.04 \times 10^{ - 4}\,\,\mathrm{N\,ms/rad},\,R_{\mathrm{a}} = 0.61\,\Omega\).

Simulation results are shown in Figs. 3, 4, 5 and 6.

## Experiments

In this section, we first give a brief introduction of the experimental setup and then present the experimental results. The parameter uncertainties, the robot velocity-related viscous and coulomb friction and nonlinearities (e.g., dead-zone and backlash) are not involved in the derived dynamic model (5). Thus, the effectiveness of the proposed controller should be verified through experiments.

### Experimental setup

*C*on the central controller. The sampling time of the control system was set as 10 ms. The Euler’s method was used as the discretization method. The same circle trajectory used in the simulation is selected as the reference trajectory in the experiment. For comparison, the controller gain is set as the same with the simulations, i.e.,

### Experimental results

## Conclusions

In this paper, a passivity-based trajectory tracking control has been proposed for an omnidirectional mobile robot. The passivity properties of the prototype robot have been analyzed. It is shown in our analysis that the prototype robot itself is an output strictly passive system and is a fully damped system. The robot itself has enough damping forces due to the large gear reduction ratio of the motors. As a result, only energy shaping (i.e., position feedback) is needed in our proposed controller. In fact, only the rotational velocity of the robot is needed. Stability analysis shows that globally asymptotic stability can be guaranteed. Both simulations and experimental results have shown the effectiveness of the proposed control design.

In the future work, we will improve the performance of the proposed control design by compensating the modeling errors and external disturbances.

## Declarations

### Authors' contributions

CR designed the control system and drafted the manuscript. YS designed the experimental system and conducted the experiments. SM supervised the study. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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