Proceedings of the 2015 IEEE Conference on Robotics and Biomimetics Zhuhai, China, December 6-9, 2015
A Gyroscope-Based Inverted Pendulum with Application to Posture Stabilization of Bicycle Vehicle Hongzhe Jin*, Decai Yang, Zhangxing Liu, Xizhe Zang, Ge Li, Yanhe Zhu
Abstract—This paper presents a new inverted pendulum system, whose core stabilizing device is designed based on gyroscopic precession effect. Traditional inverted pendulum system, such as inertia wheel pendulum, gains the required balance torque through the control over the acceleration and deceleration of inertia wheel. Considering the issues of the inefficiency, inflexibility and large capability required for motor driver of the traditional inverted pendulum system, the feature of the new inverted pendulum system possesses the advantages of simple structure, quick response, and high efficiency in control. The gyroscopic precession effect of high-speed gyroscopic rotor effectively provides a large torque to realize stable equilibrium of the new inverted pendulum system. Non-disturbance and disturbance balance experiments of the physical prototype and application of the stabilizing device to posture stabilization of bicycle vehicle are presented in this paper. The results showed that the stabilizing device responded quickly and realized stable equilibrium of the new inverted pendulum system efficiently. Besides, the consequence of application validated that the stabilizing device owns a strong anti-interference ability and a good applicability.
I. INTRODUCTION The inverted pendulum system (IPS) is a multivariable, nonlinear and static unstable system with strong couplings. Therefore it’s constantly considered as an ideal experimental platform for the research and verification of many control theories [1]-[7]. Various inverted pendulum systems with different structures, such as single-link inverted pendulum [8], double-link inverted pendulum [9], triple-link inverted pendulum [10], rotary inverted pendulum [11] and inertia wheel pendulum [12], were designed for the validation of balance control algorithms. However, since most of the inverted pendulum systems [8]-[11] except for inertia wheel pendulum system are balanced under extra driving force or driving torque from a fixed foundation or moving cart, they can only implement self-equilibrium but be applied as stabilizing devices on other mechanical platforms. Apart from research on self-equilibrium, the traditional inverted pendulum system such as inertia wheel pendulum system was widely applied to the balance control of some other objects, especially unicycle robot [13], [14]. By controlling the spin angular acceleration and deceleration of inertia wheel, the inertia wheel pendulum system can generate additional torque to balance the gravity torque and to steer the unicycle robot. Nevertheless, the main problems are Manuscript received July 25, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61004076, Grant 61473102, and Grant 61273316, in part by the Postdoctoral Science Foundation of China under Grant 20100480995. Authors are all with the School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China (corresponding author: Hongzhe Jin (e-mail:
[email protected], phone: +86-451-864 13382; fax:+86-451-86414538).
978-1-4673-9674-5/15/$31.00 © 2015 IEEE
low working efficiency and demanding high motor drive capability according to the results of our previous studies [15]–[18]. Compared with the inertia wheel pendulum system, gyroscopic stabilizing devices are more efficient, more stable and of better anti-interference ability. According to the law of conservation of angular momentum, a heavy mass spinning at very high speed provide large resistance to the interference of spin axis. Besides, by spinning a gyroscopic rotor under high speed, the gyroscopic stabilizing device can generate relative huge precession torque (balance torque) with small directional changing rate of its precession angle. Therefore, the gyroscopic stabilizing devices are of high efficiency, stability and anti-interference ability. This paper proposes a new gyroscope-based inverted pendulum with compact structure. Its core part is a stabilizing device based on gyroscopic precession effect (GPE). The stable equilibrium of the system is realized only dependent on the embedded stabilizing device. The compact and modular configuration is especially conducive to application. Besides, since the gyroscopic rotor spins in an approximately constant and high speed, little electrical energy is need for generating precession torque by changing the direction of its spin axis respective slowly. Thus, the new IPS is also efficient. Non-disturbance and disturbance balance experiments of the physical prototype was conducted and application of the stabilizing device to posture stabilization of bicycle vehicle were also presented in this paper. All the results show that the new inverted pendulum system possesses quick response and high efficiency in control. This paper is composed of four sections including the introduction. In section II, the stable equilibrium principle, physical prototype structure and dynamics model of the new IPS are described in detail. The balance experiments of non-disturbance and disturbance, and the application of the stabilizing device to posture stabilization of bicycle vehicle are presented in section III. In section IV, conclusion is given concerning the results of this work. II. STRUCTURAL DESIGN A. Stable equilibrium principle Fig. 1 illustrates the stable equilibrium principle of the stabilizing device proposed based on GPE. Where 𝐼𝑧 , 𝜔𝑧 , 𝜔𝑦 , 𝜃 and 𝜏𝑥 denote the moment of inertia about 𝑍𝑐 axis, the rotation angular velocity, the precession angular velocity, the precession angle and the precession torque of the gyroscopic rotor respectively. The correlative equation is represented by
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𝜏𝑥 = −𝐼𝑧 ∙ 𝜔𝑧 ∙ 𝜔𝑦
(1)
Fig 1. Stable equilibrium principle of stabilizing device
As expressed in equation (1), when the moment of inertia of the gyroscopic rotor about 𝑍𝑐 axis is fixed, the precession torque is proportional to the product between the rotation angular velocity and the precession angular velocity. The drawback is that the rotation angular velocity is difficult to control. Therefore, the rotation of gyroscopic rotor maintains a relatively high and stable speed driven by disc type brushless DC motor but it’s not under control and regulation. The precession angular velocity is the actual control variable. And the precession movement of gyroscopic rotor is driven by DC servo motor. An effective torque and a disturbance torque could be divided from the precession torque, calculated respectively by 𝜏𝑥 cos 𝜃 and 𝜏𝑥 sin 𝜃. And the precession angle 𝜃 changes in a range of (−𝜋/2, 𝜋/2). The effective torque reaches the largest value when 𝜃 is zero , decreases gradually as the increasing of the absolute value of 𝜃 and reaches zero when the absolute value of 𝜃 equals 𝜋/2, which means no control effect exist. In order to obtain a large effective torque, the stabilizing device should ensure that the value of the gyroscopic precession angle changes in the vicinity of zero point. The disturbance torque which varies on the contrary to the effective torque could not be eliminated availably by itself. Thus a bottom wheel with semi-cylindrical structure is developed to ease the influence of the disturbance torque, viewed in Fig.2.
The main components are the stabilizing device and the bottom wheel. The overall dimensions of the physical prototype model are 177 mm ∗123 mm ∗236 mm and the total mass is 3.525 kg, including the controller, the electronic speed governor, the motor driver and so on. Among them, the stabilizing device is composed of internal stent, external stent, transmission mechanism and gyroscopic body. The proposed structure introduced some special design and, therefore, brought many advantages, shown as follows: (1) Since gyroscopic rotor consists of only a steel turntable and a directly embedded high-speed disc type motor, the size of the gyroscopic body is reduced and the structure becomes more compact. (2) The gyroscopic body uses a design of single dual-rotor gyroscopic structure, which increases the moment of inertia of the gyroscopic body, generates a larger precession torque and lowers the required drive capability of high-speed motor. (3) As a bearing structure, the bottom wheel adopts a semi-cylindrical structure, which increases the bearing area and guarantees the new IPS swinging freely and better resistance against the impact of disturbance. C. Dynamics Model
B. Prototype structure
Fig 3. Dynamic model of the new IPS
Fig. 3 illustrates the dynamic model of the new IPS. Where σ(𝑜; 𝑥, 𝑦, 𝑧) , σ1 (𝑜𝑏 ; 𝑥𝑏 , 𝑦𝑏 , 𝑧𝑏 ) and σ2 (𝑜𝑐 ; 𝑥𝑐 , 𝑦𝑐 , 𝑧𝑐 ) denote respectively the absolute coordinate, the gyroscopic body coordinate and the gyroscopic rotor centroid coordinate. Coordinate σ1 rotates with an angular velocity 𝜔1 and the rotation angular velocity of the gyroscopic rotor is 𝜔2 . The definitions related to the dynamic model are listed in the Table I. Due to the limitation of pages, the computational procedure about the dynamical equation is skipped. According to the principle of Lagrange-d’Alembert and normalized Euler-Lagrange equation of motion, the general form of dynamical equation is represented by 𝜏 = 𝐷(𝑞)𝑞̈ + 𝐶(𝑞, 𝑞̇ )𝑞̇ + 𝐺
Fig 2. Prototype model of the new IPS
Fig. 2 displays the new IPS prototype model proposed. 2104
(2)
𝜙
TABLE I Symbol 𝜙，𝜃 𝜏𝜙 ，𝜏𝜃
MAGNETIC SYMBOLS Quantity
Gravity constant
𝑙
Center of gravity of the external stent
𝑀
Mass of the total IPS
𝑚
Mass of the two gyroscopic rotors together Mass of the internal stent and the external stent
𝜙
𝑡
𝐻
Length between the rotor centroid and the σ1 coordinates origin
𝐿0
Length between the σ2 coordinates origin and the σ1 coordinates origin
𝑘
A 0-1 constant, 𝑀𝑔𝑘𝐿0 = 𝑚𝑔𝐿0 + 𝑚𝑡𝑙𝑡 𝑔𝐿0 + 𝑚𝑤𝑧𝑗 𝑔𝑙2
𝐽0
Moment of inertia of the total IPS about X axis of the 𝜎 coordinates, 𝐽0 = 𝑚𝑡𝑙𝑡 𝐿0 2 + 𝑚𝐿0 2 + 𝑚𝑤𝑧𝑗 𝑙2
𝐽1 ，𝐽2
Moment of inertia of the two gyroscopic rotors about 𝑋𝐶 or Y𝑐 and Z𝐶 axis of the σ2 coordinates respectively.
𝐽3 ，𝐽4 ，𝐽5
Moment of inertia of the gyroscope body about 𝑋𝑏 , Y𝑏 and Z𝑏 axis of the σ1 coordinates respectively. 𝐽3 = 𝐽1 + 𝐽𝑡𝑥 + 𝑚𝐻 2 , 𝐽4 = 𝐽2 + 𝐽𝑡𝑦 , 𝐽5 = 𝐽1 + 𝑚𝐻 2 + 𝐽𝑡𝑧
𝐽𝑡𝑥 ，𝐽𝑡𝑦 ，𝐽𝑡𝑧
Moment of inertia of the internal stent about 𝑋𝑏 , Y𝑏 and Z𝑏 axis of the σ1 coordinates respectively.
𝐽𝑤𝑥 ，𝐽𝑤𝑦 ，𝐽𝑤𝑧
Moment of inertia of the external stent about 𝑋𝑏 , Y𝑏 and Z𝑏 axis of the σ1 coordinates respectively.
𝐷(𝑞)𝑒̈ + 𝐶(𝑞, 𝑞̇ )𝑒̇ + 𝐾𝑃 𝑒 = −𝐾𝐷 𝑒̇ − 𝐾𝐼 ∫𝑡 𝑒 𝑑𝑡 0
(5)
III. EXPERIMENTS AND APPLICATIONS A. Experimental Control Platform
where
Fig 4. Control structure of the new IPS
𝜏𝜙 𝜙 𝑀𝑔𝑘𝐿0 sin ϕ 𝜏 = [ 𝜏 ]; 𝑞 = [ ]; G = [ ]; 𝜃 𝜃 0 𝐽 + 𝐽3 cos 2 𝜃 + 𝐽4 sin2 𝜃 + 𝐽𝑤𝑥 0 𝐷(𝑞) = [ 0 ]; 0 𝐽5 𝑐11 𝑐12 1 ̇ 𝐶(𝑞, 𝑞̇ ) = [𝑐 0 ]; 𝑐11 = 2 (𝐽4 − 𝐽3 )𝜃 sin 2𝜃; 21 𝑐12 = −𝑐21 =
1 2
(𝐽4 − 𝐽3 )𝜙̇ sin 2θ + (𝐽2 − 𝑚𝐿0 𝐻) 𝜔cos θ.
The matrix 𝐷(𝑞) is a positive definite matrix of inertia term, the matrix 𝐶(𝑞, 𝑞̇ ) denotes the centrifugal force and Coriolis force term. The gravitational moment term of the new IPS is represented by the matrix 𝐺. Matrix 𝑞 is with respect to time 𝑡. Matrix 𝑞0 represents the initial state of the new IPS at initial time 𝑡0 and is a constant matrix. Under the assumption of 𝑞̇ 0 = 𝑞̈ 0 ≡ 0 and 𝑒 = 𝑞0 − 𝑞 where 𝑒 denotes the tracking error, the equation (2) could be turned into the following form: 𝜏 = 𝐷(𝑞)𝑒̈ + 𝐶(𝑞, 𝑞̇ )𝑒̇ + 𝐺
(3)
The PID control law based on gravitational compensation 𝐺̂ could be represented as follows: 𝑡 𝜏 = 𝐾𝑃 𝑒 + 𝐾𝐼 ∫𝑡 𝑒 𝑑𝑡 + 𝐾𝐷 𝑒̇ + 𝐺̂ 0
where
𝜙
𝐾𝑃 𝐾 𝐾 ], 𝐾𝐼 = [ 𝐼𝜃 ], 𝐾𝐷 = [ 𝐷𝜃 ]. 𝜃 𝐾𝑃 𝐾𝐼 𝐾𝐷
Matrix 𝐾𝑃 , 𝐾𝐼 and 𝐾𝐷 denote respectively the control parameters of proportion, integral and differential for PID controller. Matrix 𝐺̂ estimates the gravitational moment term of the new IPS. By adjusting the value of those parameters, a relative accurate estimation of the gravitational moment could be achieved, namely 𝐺̂ − 𝐺 = 0 . By substitute equation (3) into equation (4), it can be simplified as
Joint torqueInclination angle, precession angle
𝑔
𝑚𝑡𝑙𝑡 ，𝑚𝑤𝑧𝑗
𝐾𝑃 = [
(4)
Fig. 4 shows the control structure of the new IPS. The control structure is composed of the PLC module, power module and driver module. The PLC module connects to the PC via J-link for programming and debugging, and realizes the stabilizing device working properly. MPU6050 is a micro gyro sensor to measure the lean angle ϕ and angular velocity of inverted pendulum system. And STM32F103C8 is the processor of controller. The power module provides the required operating voltage of each module. Motor driver and speed regulation compose the driver module. In order to achieve a relative accurate control, an encoder is used to measure the working status of the DC servo motor. The DC servo motor takes the Maxon: RE25, which has a rated torque of 28.8 mNm connected to a gearbox with a reduction ratio of 84:1. The Disc type brushless DC motor takes the DYS: BE3608-630kv, which has a peak revolving speed of 10335 rpm. Disc type brushless DC motor drives the gyroscopic rotor in a stable high-speed rotation, and DC servo motor is adopted to drive the precession of the gyroscopic rotor. Table II gives the estimation of the structure parameters of the IPS. The maximum torque produced by the GPE of the rotors reaches 7895.7 mNm, and is greater than the maximum estimation of gravitational torque, which is 7000 mNm. The side balance control of the new IPS is simplified into the control of the DC servo motor, and the speed of DC servo motor is the actual control output. Through the transmission mechanism, the DC servo motor speed transforms into the
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precession angular velocity 𝜔1 of the gyroscopic rotor. On the basis of GPE, the lean angle 𝜙 of the new IPS is adjusted by the effective torque, which is determined by the precession angle 𝜃 and the precession angular velocity. When lean angle 𝜙 changes in a small range near zero, the new IPS achieves stable equilibrium. As expressed by the equation (5), PID control algorithm is adopted and control parameters are 𝜙 𝜙 𝜙 𝐾𝑃 , 𝐾𝐼 , 𝐾𝐷 , 𝐾𝑃𝜃 , 𝐾𝐼𝜃 and 𝐾𝐷𝜃 . The initial values of the control parameters are determined by the results of simulation. By constantly adjusting the values of the control parameters in experiments, the observed curves of the control parameters are consistent with the target curves commendably. Table III shows a group of feasible values of the control parameters, which realize the stable equilibrium of the new IPS successfully.
control effect would be for the PID control algorithm. The regulating phase represents the fine-tune time of the lean angle, so as to maintain stable equilibrium preferably. The phenomenon of the non-disturbance balance experiment indicates that the stabilizing device designed on the basis of GPE through PID control algorithm, is able to realize stable equilibrium. The experimental data are to be analyzed as follows.
TABLE II THE ESTIMATION OF THE STRUCTURE PARAMETERS
(a)
OF THE IPS
Mass of Single Rotor (kg) 0.4 Max deflection angle of rotor (rad) ±(π/3)
Distance from Center of Rotor to Ground (m) 0.2 The moment of inertia of two rotors (kg·m2) 1.0×10-3
Rotational Speed of Rotor (rpm) 6000 Mass of gyroscopic body (kg) 1.8
Max Angular Speed of Rotor Precession (rpm) 120 Total mass of the IPS (kg) 3.5 (b)
TABLE III VALUES OF PID CONTROL PARAMETERS 𝜙
𝜙
𝜙
Parameters
𝐾𝐼
𝐾𝐷
𝐾𝑃𝜃
𝐾𝐼𝜃
𝐾𝐷𝜃
𝐾𝑃
Value
5
6
2
0.005
0.3
40
B. Non-disturbance Balance Experiment The experiments of the stabilizing device can be divided into two parts: non-disturbance balance experiment and disturbance balance experiment. The non-disturbance balance experiment is to validate the quick response and convergent property of the new IPS. The robustness and the ability to resist outside disturbance of the new IPS are clarified in the disturbance balance experiment.
(c)
(d) Fig 6. Response curves of non-disturbance balance experiment
Fig 5. Process of non-disturbance balance experiment
Fig. 5 reveals the steer process of the new GPE stating from a certain lean angle. It could be divided into four phases: the initial phase (a), the oscillation phase (b), the regulating phase (c) and the equilibrium phase (d). The initial lean angle should not be set too large, generally less than 20°, to prevent the balance control from failure. So it is set at 17°in this experiment. The oscillation phase is a rapid response period and the shorter time is required, the better
Fig. 6 illustrates the response curves of lean angle 𝜙, precession angle 𝜃 and their velocities in the non-disturbance balance experiment. According to Fig. 6, after about four oscillations with 1.6s, the curves of both angles turn to be convergent and stable with small fluctuations, near zero. The total process can be divided into four phases according to the time. The initial phase is at 0s, oscillation phase is between 0~1.6s, regulating phase is between 1.6~3s and equilibrium phase is after 3s. Fig. 6 (a) and (b) present the response curves of lean angle and its velocity. The maximum oscillation amplitude of 𝜙 is 6.27°and the fluctuation amplitude is 0.2°. The larges value of its velocity is 54.7deg/sec at 0.62s, and turns to 1.5deg/sec at equilibrium phase. The two curves converge to zero rapidly after a brief oscillation phase, clarifying the good convergence of the control system.
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Fluctuation of the curve within a small amplitude range while it maintains stable equilibrium indicates excellent stability of the control system. The response cure of precession angle is showed at Fig. 6 (c). The maximum oscillation amplitude of 𝜃 is 30°and when at equilibrium phase, it turns to 0.5°. Fig.6 (d) presents the comparison between the given theoretical control curve and the response curve observed of the precession angular velocity. By contrast, certain differences of phase and amplitude are existed. The phase of the theoretical control curve advances about 0.1s than the response curve observed, due to the system response delay. C. Disturbance Balance Experiment The disturbance balance experiment presents that the new IPS regains stable equilibrium from disequilibrium, after the impact of outside disturbance. Derived from unicycle robot, the new IPS would encounter inevitably numerous external disturbances. So the disturbance balance experiment for the new IPS is necessary. The process of the disturbance balance experiment can be also divided into four phases: the initial phase, the oscillation phase, the conditioning phase and the equilibrium phase.
(a)
Fig. 7 shows the response curves of the lean angle ϕ, the precession angle θ and their velocities in the disturbance balance experiment. Each oscillation period of the curves means once outside disturbance acts on the new IPS. The time needed to regain stable equilibrium depends on the features of disturbance, such as the force, the time and the position. But the process still can be divided into the four phases as mentioned above. In this experiment, there are five times disturbances occurred at 3.2s, 4.5s, 10.2s, 17.3s and 20.5s respectively. Fig. 7 (a) and (b) present the response curves of the lean angle 𝜙 and its velocity, respectively. The maximum oscillation amplitude of 𝜙 is 3°, which means the disturbance is bearable for the stabilizing device. It reveals that the stabilizing device has an excellent anti-inference ability. In Fig. 7 (b), the largest value of its velocity is 46.5deg/sec. These two curves respond quickly and achieve convergence after a short period of oscillation, which costs 0.5s, 1.4s, 1.1s, 1.5s and 1.5s respectively. It indicates that the stabilizing device has an excellent regulating ability, and the new IPS could regain stable equilibrium in a short time after a disturbance. Fig. 7 (c) and (d) show the response curve of the precession angle and the contrast between the response curve observed and the given theoretical control curve of the precession angular velocity. At Fig. 7 (c), the maximum oscillation amplitude of θ is 26°. The precession angle responds quickly and soon converges, but convergence point varies in a small range and owns a zero drift. As same as the non-disturbance balance experiment, there exists certain differences of phase and amplitude between the given theoretical control curve and the response curve, shown in Fig.7 (d) . The phase of the theoretical control curve advances about 0.1s than the response curve observed. In order to get a better control effect, the values of control parameters should be adjusted through multitudinous experiments. Shown from the two experiments, the new IPS can bear disturbances that can cause a lean angle less than 17°. So, the stabilizing device based on GPE has a good regulating and anti-inference ability to keep the new IPS stable equilibrium. D. Application
(b)
(c)
(d) Fig 7. Response curves of disturbance balance experiment
Fig 8. Process of application to posture stabilization of bicycle vehicle
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Fig. 8 shows the application of the new IPS to posture stabilization of bicycle vehicle. During the balance experiments of non-disturbance and disturbance, the new IPS does not move back and forth and thus the application of the stabilizing device to posture stabilization of bicycle vehicle is proposed. The structure of bicycle vehicle is composed of the stabilizing device and a platform with two free wheels. The bottom wheel of the original IPS is inoperative and the later bicycle vehicle could still swing freely. Without the working stabilizing device, the bicycle vehicle could not stand steadily. The balance experiments are conducted on an inclined floor platform.
characteristics of quick response, fast convergence, strong regulation and certain anti-inference ability. Through the application of the stabilizing device to posture stabilization of bicycle vehicle, the dynamic regulating ability and practical platform applicability of the stabilizing device were validated. However, for the sake of promoting and enhancing the side balance control effect of the stabilizing device to adapt it to some more complex environments, a lot of works still need to be done. REFERENCES [1] [2] [3]
[4] [5]
(a)
[6] [7]
[8] [9]
(b)
[10]
Fig 9. Response curves of application to posture stabilization of bicycle vehicle
Fig. 9 reveals the data analysis of the application, which would not be illustrated in detail in order to avoid tautology. In order to simulate the possible behaviors of the bicycle vehicle under dynamic environment, three times disturbances are introduce into the experiments, which cause the three times oscillations shown in Fig. 9. As a result, the oscillation amplitudes of lean angle and precession angle are less than 3° and 23°, respectively. The results validated the high efficiency and anti-interference of the new IPS device. In the application, with the installation of stabilizing device, the bicycle vehicle can maintain stable equilibrium during traveling commendably which shows the stabilizing device possesses good dynamic characteristics and extensive practical application prospects.
[11]
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IV. CONCLUSION This paper presents a stabilizing device based on gyroscopic precession effect and achieves stable equilibrium of the new inverted pendulum system. The dynamical equation and efficient PID control algorithm are proposed. The balance experimental results of non-disturbance and disturbance verify that the stabilizing device owns the
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