Dynamical model of a new type of self-balancing tractor-trailer-bicycle

Abstract. In this paper, we focused on a Self-Balancing Tractor-Trailer-Bicycle(TTB) and developed an under-actuated dynamical model for the system. The bicycle is characterized with two parts, that is a tractor and a trailer, and considering the nonholonomic constrains from no-slipping contacts of its three wheels and the flat ground, we presented a dynamical model for the bicycle by using Chaplygin equation. The model suggest that the TTB should be an under-actuated system with three DOF (degree of freedom) and there are two driving-torque inputs. An inverse dynamics and a virtual prototype simulations are given to demonstrate the correctness of the proposed dynamical model.


Introduction
Self-balancing bicycle is the combination of bicycle mechanism and balance control technology. For this kind of two-wheeled mechanism, on the one hand, it can satisfy ones' needs of convenient travelling and labor-saving due to its lightweight and flexible body; on the other hand, the bicycle riders hope in some case that it can balance automatically, which can get rid of the dependence on ones' "driving". So far, the research on the self-balancing bicycle can be grouped into two types: "without mechanical regulator" and "with mechanical regulator".
Researchers who focus on self-balancing bicycle without mechanical regulators include Jones [1], Tanaka [2], Kooijman [3], Huang [4,5], and Li [6], etc. These researchers believed that the unmanned bicycle can achieve the dynamic balance of the body by governing the handlebar turning and the wheels running without adding additional mechanical regulators.
In the literatures [4][5], Huang introduced the principle of instantaneous rotation axis to analyze the constraints of bicycle robots, and used Lagrange method to establish its dynamic model. Huang also designed a motion controller based on partial feedback linearization method, and finally gave some physical prototype experiments, e.g., in situ, circular motion, linear balanced walking, etc.
Researchers who study self-balancing bicycles with mechanical regulators include Lee [7], Bui [8], Liu [9], Jin [10], Yin [11], Kim [12], etc. These researchers designed mechanical adjusting devices of moving or rotating for their bicycles. They stressed the dominant role of the mechanical adjusting devices in maintaining the balance of the body. The balance adjustment mechanism includes inertial flywheel (or rotating rod, or pendulum rod), translation mass block, mechanical gyro, etc.
At present, the research of self-balancing bicycle is mostly focused on the self-balancing of the body, and few of them can pay attention to the problem of the payload capacity of the system. Since the self-balancing bicycle without mechanical regulator has a simpler structure, fewer driving motors, lighter weight, and of more energy-saving. The existing self-balancing bicycles usually adopt the narrow structure of two wheels arranged back and forth, and the wheelbase between their two wheels has an important impact on their balance performance. If the wheelbase increased for improving the load-carrying capacity, it may cause many unexpected problems aroused from the frame deformation and flexibility reduction, and it is easy to lead the bicycle to lose balance.
How to improve the payload capacity without reducing the balance performance of the system? Inspired by the multi-section train, we proposed a new self-balancing bicycle mechanism composed of two-wheel tractor and single-wheel trailer. This kind of two-section bicycle retains the advantages of the traditional bicycle body in structure. If a breakthrough can be made in balance theory and experiment, this kind of mechanism should become a new type of convenient road traffic tool.
The structure of a new tractor-trailer-bicycle (TTB) are described in details in this article, and its dynamic model of the system was developed seriously by using Chaplygin equation.

Mechanical structure
The TTB consists of roughly two parts: a tractor and a trailer, which is shown in Fig. 1~Fig  As it is seen in Fig. 1~Fig. 2, the tractor includes a tractor frame, a handlebar, a front wheel of the tractor and a rear wheel. Both the handlebar and the rear wheel of the tractor are rotated around the frame, and the front wheel of the tractor rotates around the handlebar. The trailer comprises the front fork of the U-shaped block, the rear fork of the U-shaped block and the wheel. The front fork of the U-shaped block can be rotated up and down around the tractor frame, the rear fork of the U-shaped block can be rotated around the front fork of the U-shaped block, and the wheel of the trailer is rotated around the rear fork of the U-shaped block.

Constraint analysis
We suppose the bicycle was running on a flat plane, then the angular velocity of 1 B can be given as: , their angular velocity should be calculated as: where i q  ( 4,5,8,9 i = ) denotes the angular rate of i B ( 2,3, 6, 7 i = ), respectively. Assume that the bicycle would not slide on the horizontal plane, so the speed of 1 P and 2 P is zero, then there we can get the following equation: In addition, there are the following equations: (2) (1) In (1)~ (5) and (8)~(11), we can get the following four nonholonomic constraint equations: Similarly, the speed of 3 P is zero, so we can get: In (12) ). In addition, considering the geometric constraints, we can get the following equations: Eventually, we can derive the following two holonomic constraint equations: We assume that the bicycle is running on a flat ground. The attitude matrix of the front fork of the U-shaped block is denoted by α R , and the roll angle is denoted by α , and the attitude matrix of the rear fork of the U-shaped block is denoted by β R , and the roll angle is denoted by β . As a result, we would get two equations :

Velocities of the COM
We set the velocity of the geometric center of 4 B as: Considering the principle of the relative motion, we can get the velocity of k B ( 1, 2,3,5, 6, 7 k = ), respectively, as follows: ( ) where (1) Ci v is the velocity of i B ( 1, 2,3, 6, 7 i = )in{1}, (5) 6 o l is the position vector in{5} from the origin of {6}to the origin of {5}.

Kinetic energy and potential energy
According to the derived 1, 4 i k = =  ) in Eq. 22~Eq. 28, we can calculate the system's kinetic energy as: , respectively. By substituting the nonholonomic constrains Eq. 12~Eq. 17 into T , we will get another form of the kinetic energy T  . Simultaneously, system gravity potentialU can be given as:

Dynamical model
Considering the following form of Chaplygin equation: , , where T is the kinetic energy and T  is the kinetic energy by substituting nonholonomic constrains into T ;

Model verification
We will demonstrate the reliability of the model (see Eq. 39) by two different approaches. One is the use of an inverse dynamic simulation of Eq.39 under a given balanced trajectory in Matlab, from which we compare the energy increment with the input work of the dynamic bicycle. The other is the use of a virtual prototype simulation in ADAMS, by which we compare the driving torque of the handlebar in Adams with the model-calculated handlebar driving torque. Table 1 shows the physical parameters which would be used in the numerical simulation. Note that we obtain the parameters from the measurement of a virtual TTB prototype in Solidworks.

Inverse dynamics simulations
The simulation is perform with two steps.
Here, 1 Fig. 3 examines two kinds of kinetic energy: T1 is get by the current velocity and T2 is get by the previous velocity and the elementary work. Fig. 4 shows the difference between the mechanical energy and the work of the running bicycle.  As seen in Fig. 3~Fig. 4, while calculated by use of different variable, the two kinetic energy are coincident, and the difference between the increment of the mechanical energy and the elementary work is less than -3 10 (see Fig. 4). The results show that our dynamical model (Eq. 39) strictly obey the law of conservation of energy.

Virtual prototype simulation
The simulation is as follows:  Experiment description First, we build the virtual prototype in Adams platform, and add kinematic pairs and constraints to each rigid body part. Secondly, we define the type of contact between the wheels of the TTB and the ground, and add static friction and dynamic friction. The parameter settings such as system quality, moment of inertia, length and body structure length required for simulation are shown in Table 1.Finally, the angular velocity of the 3 B is set to 650r/min, then a simply PD controller is designed as: The handlebar is governed by the controller to balance the TTB. The simulation continue with 20s due to the space limit. The relative data of the virtual prototype are exported for the post process after the complement of the simulation. 

Experiments result and analysis
We calculated the driving torque of the handlebar 2 B through the dynamical model in Eq.
39. Also, we got the measurement of this torque from ADAMS. Fig. 5 show that TTB is running on a flat plane in the ADAMS simulation platform environment.    It is illustrated that the two kinds of the driving torque of the handlebar 2 B exhibited the similar trend with a little difference in the amplitude. The reasons for the difference maybe as follows: 1) There definitely is wheel slippage because the angular acceleration of the front wheel of the tractor 5 B is relatively large form 0 to 0.5 second; 2) The measurement errors of virtual sensors; 3) The structural parameter measurement errors; 4) The proposed dynamical model is developed under ideal assumption without considering the friction between the wheels and the ground etc. As a conclusion, the results of the two simulations seriously demonstrate the correctness and the reliability of the proposed dynamical model (Eq. 39).

Conclusions and future work
One of the contribution of this research is that we suggest that the TTB can be controlled balance by the handlebar. Another contribution of the research might be that we explored the dynamical model for the system. Our model illustrates that the TTB is explicitly a nonholonomic and under-actuated system, which consist of three independent velocities and two control-torque inputs. With the comparison between numerical and virtual prototype simulation, we validated that the reliability of our dynamical model. However, by so far, there is lack of realistic test to provide further support for our theoretical analysis; so, our next work should concentrate on the physical experiments.