Analysis and decoupling control of a permanent magnet spherical actuator Liang Zhang, Weihai Chen, Jingmeng Liu, and Xingming Wu Citation: Review of Scientific Instruments 84, 125001 (2013); doi: 10.1063/1.4833681 View online: http://dx.doi.org/10.1063/1.4833681 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/84/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A control algorithm for the deepwind floating vertical-axis wind turbine J. Renewable Sustainable Energy 5, 063136 (2013); 10.1063/1.4854675 Optimal design of stator interior permanent magnet machine based on finite element analysis J. Appl. Phys. 105, 07F104 (2009); 10.1063/1.3067486 Real-time optimal torque control of fault-tolerant permanent magnet brushless machines J. Appl. Phys. 97, 10N510 (2005); 10.1063/1.1852439 A comparative analysis of permanent magnet-type bearingless synchronous motors for fully magnetically levitated rotors J. Appl. Phys. 83, 7121 (1998); 10.1063/1.367540 Analysis of a spherical permanent magnet actuator J. Appl. Phys. 81, 4266 (1997); 10.1063/1.364797
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125001 (2013)
Analysis and decoupling control of a permanent magnet spherical actuator Liang Zhang, Weihai Chen,a) Jingmeng Liu, and Xingming Wu School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
(Received 17 July 2013; accepted 12 November 2013; published online 5 December 2013) This paper presents the analysis and decoupling control of a spherical actuator, which is capable of performing three degree-of-freedom motion in one joint. The proposed actuator consists of a rotor with multiple PM (Permanent Magnet) poles in a circle and a stator with circumferential coils in three layers. Based on this actuator design, a decoupling control approach is developed. Unlike existing control methods that each coil is responsible for both the spinning and tilting motion, the proposed control strategy specifies the function of each coil. Specifically, the spinning motion is governed by the middle layer coils with a step control approach; while the tilting motion is regulated by upper and lower coils with a computed torque control method. Experiments have been conducted on the prototype to verify the validity of the design procedure, and the experimental results demonstrate the effectiveness of the analysis and control strategy. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4833681] I. INTRODUCTION
Many industrial applications such as manufacturing, helicopters or underwater vehicles, automobile wheel, and robotic joints require orientation control of the rotating shaft.1 Conventionally, this type motion is achieved by connecting a few single-axis actuators in series or in parallel with external mechanism. However, this combined actuation system has intrinsic disadvantages such as bulky structure, large backlash, slow dynamic response, singularity existence in workspace, and lack of dexterity.2 To overcome these drawbacks, the researchers have proposed the concept of spherical actuator, which can realize three-DOF (degree-of-freedom) motion in one joint. Currently, several different driving techniques such as the mechanical methods, piezoelectricity, and electromagnetic forces have been proposed for the spherical actuators.3–6 Electromagnetic actuation principle has been paid more attention among them.6 The earliest spherical induction motor is proposed by Williams et al.7 It can generate 2-DOF motion in its workspace. After that, especially in recent decades, various designs and control methods have been proposed.8–18 Chen et al. have made a spherical actuator consisting of a rotor with eight PM (Permanent Magnet) poles and a stator with 24 aircore coils. The magnetic field and torque model have been analyzed, but the control performance is not included.8, 9 Lee et al. have proposed a design concept of the spherical wheel motor,10 a similar design has been developed by Kang et al.11 Its open loop control method has been proposed, while the square wave used for the spinning motion results in torque ripple and deteriorates the control accuracy.12 Rossini et al. have presented the open-loop control of a reaction spherical actuator, which can be used for the satellite attitude control.13 Öner has proposed a four pole spherical rotor, and its magnetic field and open loop control method have been analyzed.14 Maeda a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2013/84(12)/125001/10/$30.00
et al. have developed a spherical actuator consists of a rotor with four interior magnets and two sets of stators with six magnetic poles. The feed-back control has been studied, while the control performance is affected by the accumulated orientation measurement error.15 Kahlen et al. have developed a torque control method for the spherical motor, which consists of a rotor with 112 PM poles and stator with 96 coils.16 In conclusion, one common feature of the existing spherical actuators is that many coils are mounted on the stator in 3D space.19 If a certain coil is used to generate 3-DOF rotational motion at the same time, the required current input changes of spinning motion will inevitably influence the motion in the tilting direction.12 Therefore, the control accuracy of the spherical actuator will be affected. In this paper, we will present an alternative control strategy for the spherical actuator. Unlike the existing energizing strategy that each coil governs all the 3-DOF rotations, this approach specifies the function of each coil. Specifically, the coils are evenly grouped into three layers, the middle layer coils are used for the spinning motion, and the upper and the lower layer coils are used for the tilting motion. By doing so, the control of the spherical actuator becomes easier, and the effect of the coupling between the tilting and spinning motion can be reduced. In addition, since one more layer coils are assembled in the inner space of the stator, the force density of this actuator is better than the spherical actuator with two layer coils. The control method presented in this paper can decouple the tilt control of the rotor from its spinning motion. The spinning motion is controlled with step control method, and sine wave is used to reduce the oscillation and the undesired torque ripple. The tilting motion is controlled with computed torque control method, which is a model-based dynamic control method. The reminder of the paper is organized as follows. First, the structure of the spherical actuator is described. Second we analyze the dynamic and torque characteristics. After that, the control strategies for the spinning and tilting motion are designed. Then we illustrate the prototype of the proposed spherical actuator and its experimental platform, on which the
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Rotor shaft Stator
Rotor
Passive spherical joint
Base
(a)
(b)
FIG. 1. Spherical actuator. (a) CAD model of spherical actuator. (b) Assembly diagram.
experimental studies are carried out. Finally, the paper is summarized in conclusions. II. STRUCTURE DESCRIPTION
Compared to the conventional 1-DOF actuators, the design of a 3-DOF spherical actuator is quite different. Fig. 1(a) shows the CAD (Computer Aided Design) model of the proposed spherical actuator, which consists of a rotor housed in a spherical shell shaped stator. Fig. 1(b) shows the assembly diagram of the spherical actuator. A. Rotor
Fig. 2 shows the structure of the rotor. Eight PM poles are uniformly assembled on the equatorial plane of the rotor. The unit vector on the magnetization axis of PM poles in the rotor frame is given by ri = (−1)
i−1
[ cos ϕi
sin ϕi
T
0] ,
i = 1, 2, . . . , m, (1)
where ϕ are the azimuth angle of the PM pole and m is the number of PM poles. To reduce the inertial of the rotor, eight evenly distributed trapezoid holes are processed on the rotor shell. The rotor is supported and measured by the passive spherical joint (Fig. 3). This spherical joint is composed of a 1-DOF passive rotary joint in conjunction with a 2-DOF passive universal joint. An encoder and a 2-axes tilt sensor are
FIG. 2. Rotor structure.
FIG. 3. Passive spherical joint.
assembled on the passive spherical joint. The spinning angle of the rotor can be detected by the encoder, and the tilting angle can be measured by the 2-axes tilt sensor. With this measurement method, a closed-loop control can be implemented, and thus the controllable of the actuator can be effectively improved. B. Stator
Fig. 4 shows the exploded view of the stator. Three layer holes are distributed on the stator shell and 30 coils are mounted in these holes. The middle layer coils are evenly mounted on the equatorial plane of the stator, and the upper
FIG. 4. Stator structure.
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FIG. 5. Operating principle. (a) Spinning motion. (b) Tilting motion.
and lower coils are symmetrically assembled respect to the stator equatorial plane. Because of this design, the proposed decoupling control strategy can be implemented. Moreover, because one more layer coils are distributed on the stator, the inner space of the stator is better used. Thus, the torque output ability is improved. The unit vector on the coils axis in the stator frame is given by sj = [cos ηj cos φj j = 1, 2, . . . , n,
cos ηj sin φj
sin ηj ]T ,
(2)
where η and φ are the polar and azimuth angle of the coil axis, n is the number of coils. C. Operating principle
Spherical actuator can realize three-DOF motion in one single joint. Here, the spinning motion is controlled by the middle layer coils, and the tilting motion is controlled by the upper and lower coils. Current-carrying coils act like a current controlled magnet and establish a magnetic field around it. The force is generated as a result of the interaction between the stator magnetic field and the rotor magnetic field. The symbols “N” and “S” in Fig. 5 indicate the north and south polarities, respectively. The operation principle of the spinning motion is similar to the stepper motor. Ten middle layer coils are divided into five phases to control the spinning motion. The rotor has four north and four south pole permanent magnets. The polarity of the coils depends on the current input. If the coils are energized as shown in Fig. 5(a), an anticlockwise rotation along the rotor shaft can be generated. If the current inputs of these ten coils are changed in sequence as a stepper motor, a continuous spinning motion can be generated. A change in current input from one state to another creates a change in the rotor position. If the current is not change, the rotor stays in that stable position. The operating principle of the two-DOF tilting motion is shown in Fig. 5(b). The rotor will tilt around the axis normal to the plan under the tilting torque if the coils are energized as shown in Fig. 5(b). The current inputs are regulated according to the desired rotor orientation and tilting torque. Any changes of the current inputs will result in a differential
torque that drives the rotor to its equilibrium. By varying the current inputs based on the control model, the desired tilting motion can be realized. If we energize all the three layer coils at the same time, three-DOF rotational motion can be generated. The torque model of the spherical actuator is orientation dependent, and tilting control is based on the torque model. Thus, when we control the spinning and tilting motion at the same time, the orientation changes from the spinning motion should be included in the tilting control algorithm. In our control model, the torque computation is involved in the tilting control law, and the orientation information can be obtained by the passive spherical joint. Therefore, by regulate the current inputs of all the coils under the control law, the desirable three-DOF rotational motion in the workspace can be achieved. III. DYNAMIC AND TORQUE ANALYSIS A. Rotor dynamics
XYZ Euler angles (α, β, γ ) are used to express the rotor orientation, which has no singularities in the workspace of the actuator.20 xr yr zr and x0 y0 z0 are coordinates on the rotor and stator, respectively. The zr -axis of the rotor is pointing along the rotor shaft. The rotor is a single rigid body mechanical system. Its dynamical equation derived from the Lagrange’s equation in terms of Euler angle (α, β, γ ) is given by ˙ q˙ = τ c − τ ext , M(q)q¨ + C(q, q) ⎡ J1 c2 βc2 γ +J2 c2 βs 2 γ +J3 s 2 β
where M(q) = ⎣
(J1 −J2 )cβcγ sγ J3 sβ
(J1 −J2 )cβcγ sγ
J1 s 2 γ +J2 c2 γ
⎡
0
(3) ⎤
J3 sβ 0
⎤
⎦
J3
C11 C12 C13 ˙ = ⎣ C21 C22 C23 ⎦ is the is the inertial matrix; C(q, q) C31 C32 C33 Coriolis matrix; q = [ α β γ ]T is the Euler angle vector; T τ c = [τα τβ τγ ] is the control torque; τ ext is the external torque including the load and disturbances; and J1 = Jxx , J2 = Jyy , J3 = Jzz are the principle inertial moments of the ro˙ are given tor. The elements of the Coriolis matrix C(q, q) in the Appendix. It can be seen from Eq. (3) that the rotor
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dynamics has nonlinear characteristic, many interaxis coupling terms exist in the dynamic model.
B. Torque calculation
The torque here is analyzed using the finite-element (FE) method. The FE model is established in Maxwell 3D. Since the coils are air-core, the torque has a linear property with respect to the current input, and the torque generated by all the coils can be calculated by summing up each individual coils.9 The total torque for all the coils of the actuator is T = [ Tx Ty Tz ]T = GI,
(4)
where the torque matrix G = [ G1 . . . Gj . . . GN ], Gj ∈ 3×1 and the current input I = [ I1 . . . Ij . . . In ]T . Gj is the torque coefficient, which describes the torque contribution of the j th coil. n is the number of the coils, and Ij is the current input of the jth coil. Gj is given as ⎧ m ⎨ (−1)i−1 f (ϕ )(r × s /r × s ), if r × s = 0 ij i j i j i j Gj = i=1 ⎩ 0, if ri × sj = 0, (5) where f(ϕ ij ) is the torque constant function derived by curve fitting the torque between one PM pole and one coil, ϕ ij = cos −1 (ri · sj ) is the separation angle between the PM pole and the coil. It can be seen from Eqs. (4) and (5) that the torque output of the spherical actuator is orientation dependent. IV. CONTROL DESIGN OF THE SPHERICAL ACTUATOR
A decoupling control method which decouples the tilt control of the rotor from its spinning motion is proposed. The spinning motion is controlled with step control method by the middle layer coils, and the tilting motion is controlled with computed torque control law by the upper and lower lay coils. Since that if one coil is used to regulate both the spinning and tilting motion at the same time, the demanded current changes in spinning motion will inevitably affect the motion in tilt direction. Thus, the function of each coil is specified. Such design can not only makes the control easier but also reduce the coupling between the motions in different direction.
A. Spinning motion control
The spinning motion control here is similar to the stepper motor, a given coil excitation state defines a rotor position as a result of the electromagnetic force between the stator coils and the PM poles. The diametrically opposite coils are arranged in one phase and connected in series. Ten middle layer coils are divided into five phases as Phase 1 : C1 , C6 ;
Phase 2 : C2 , C7 ;
Phase 4 : C4 , C9 ;
Phase 5 : C5 , C10 ;
Phase 3 : C3 , C8 ;
where Cj indicates the jth coil (each middle layer coil is given a label as shown in Fig. 5). The minimum step size is 9◦ , which is determined by the number of PM poles and middle layer coils. To reduce the oscillation and the undesired torque ripple, sine wave is used for the spinning motion instead of square wave. The sine wave current input does not change in sudden jumps, but instead it is smoothly changed. This will bring advantages as smoother motion and electronically controlled finer step sizes. In addition, the step-loss problem can be reduced. The current input of the jth phase is given as
4 (6) Isj = A sin ωc t + θ0 − π (j − 1) , 1 ≤ j ≤ 5, 5 where A and ωc are the amplitude and frequency of the sine wave, θ 0 indicates the initial position of the rotor in the spinning direction. The spinning period of the spherical actuator can be derived from the current input, which is given as ts = 8π /ωc . Thus, by controlling the frequency of the sine wave, the spinning rate can be regulated. The step loss can be detected by the encoder and will be compensated when necessary. B. Tilting motion control
Tilting motion is a distinct feature of the spherical actuator. The control task here is to determine the required current inputs that cause the rotor to execute a desired tilting motion. The question is divided into two parts: first, determining the applied driving torque that drives the rotor to track a desired tilting motion; then, the determination of the optimal current inputs that generates the desired driving torque. 1. Computed torque control for tilting motion
The objective of the dynamic control algorithm is to determine the driving torque that actuates the rotor to follow the commanded motion. Considering the dynamic characteristic of the spherical actuator, computed torque control method is utilized for the tilting control of the spherical actuator.21 The desired driving toque Tt = [ Tt1 Tt2 0 ]T which actuates the rotor from its initial equilibrium state to the final state under the computed torque control method is given by ˙ q, ˙ Tt = M(q)(q¨ d − Kv e˙ − Kp e) + C(q, q)
(7)
where q = [ α β 0 ] express the tilting orientation of the rotor and qd is the desired tilting orientation; e(t) = q(t) ˙ − q˙ d (t) define the tracking error and − qd (t) and e˙ (t) = q(t) its derivative, respectively; Kv and Kp are positive definite differential and proportional coefficient matrices, respectively. When substituting Eq. (7) into Eq. (3), the error dynamics can be written as T
M(q)(¨e + Kv e˙ + Kp e) = 0.
(8)
Since M(q) is positive definite, we have e¨ + Kv e˙ + Kp e = 0.
(9)
Equation (9) is a linear differential equation, which regulates the error between the desired and actual trajectories. Since
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FIG. 6. Prototype of the spherical actuator. (a) Spherical actuator. (b) Rotor. (c) Passive spherical joint.
Eq. (9) is linear, the positive definite gain matrices Kv and Kp are easy to choose to ensure the stability of the system. In addition, it can be deduced that e → 0 exponentially as t → ∞. 2. Current inputs control
The spherical actuator has more independent coils than the dimensional of the tilting torque. The redundant in the current inputs implies that an optimal control can be conducted. Here, the optimal current inputs of the upper and lower coils are obtained by minimizing the energy consumption, which is given by It = GT (GGT )−1 Tt .
(10)
If the calculated current inputs are smaller than the current limits, they can be used to energize the upper and lower coils directly, and the power consumption is minimized. However, when the calculated current inputs exceed the current limits, a fault tolerant control should be implemented to ensure the stability of the system.22 V. PROTOTYPE AND EXPERIMENTS A. Prototype
Fig. 6 shows the prototype of the proposed spherical actuator. The structure specifications are given in Table I. The material of the rotor and stator shell is aluminum, and the nonmetal material can be used in future to eliminate the eddy current effect. The working range of the tilting motion is ±15◦ , which can be increased by distributing more layer coils on the
FIG. 7. Experimental platform.
stator. The type of the encoder is Angtron-RE-20-485, which is a 14 bit absolute sensor. The type of 2-axes tilting sensor is SANG 1000, whose accuracy is up to 0.01◦ . The power consumption of the spherical actuator is about 135 W. Two cylindrical holes separated 90◦ apart that point to the stator center are designed at the equator plane of the stator shell. Similarly, two holes separated 90◦ apart are drilled at the equator plane of the rotor shell. Two centering pins can be inserted to these holes to ensure the concentricity between rotor and stator. B. Experiments
1. Experiment setup
To demonstrate the performance of the spherical actuator, an experimental platform consists of the research prototype and its control system is developed as shown in Fig. 7. The whole control system comprises two parts: the host computer and the current controller, they communicate with each other through the serial port. The host computer is mainly responsible for the control algorithm computation and communication with the current controller. A graphical user interface (GUI) program written with VC++ is developed on the host computer in the VS2010 environment, through which the control command can be sent to the host computer. In addition, the system information can
TABLE I. Structure specifications of the spherical actuator. Rotor Number of PM poles Inner/outer rotor shell Cylindrical PM pole Magnetization Moment of rotor inertia (kg/m2 × 10−3 ) Stator Number of coils Inner/outer stator shell Cylindrical coil Winding parameters
8 PM poles in one layers 53/58.5 mm Radius = 12.5 mm, height = 20 mm Axial magnetization, Br = 1.21 tesla J1 = 2.219, J2 = 2.176, J3 = 2.256
30 coils in three layer 86/98 mm Radius = 11 mm, height = 25 mm AWG 26 copper wire, 950 turns FIG. 8. Diagram of current controller.
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FIG. 9. Spinning motion tracking.
Rev. Sci. Instrum. 84, 125001 (2013)
FIG. 10. Spinning motion at low velocity.
FIG. 11. Step response of tilting control. (a) Euler angle α. (b) Euler angle β. (c) Steady state tracking error of α and β.
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FIG. 12. Tilting control of spinning rotor with three layer coils. (a) Tilting motion. (b) Spinning motion. (c) Steady state tracking error of α and β.
be displayed in real time on it, such as the orientation, current, temperature, spinning velocity, etc. According to the control command, the host computer does the algorithm calculation and sends the corresponding command to the current controller through the serial port. The current controller is in charge of providing multichannel, bipolar current inputs to the stator coils; sampling and processing the orientation and current information; sending the information to the host computer for motion control and real-time display. As shown in Fig. 8, the current controller system consists of two parts: the core control part and the power amplifier output module. The core control part is comprised of ARM (Advanced RISC Machines) and CPLD (Complex Programmable Logic Device). The ARM7 microprocessor is mainly in charge of task scheduling, and the CPLD is mainly responsible for driver function programming. The power amplifier output module is responsible for providing the current to the coils. Since the input current should be multi-channel and bipolar, the D/A chip AD5370 is utilized here. It could offer digital-to-analog converting with 40 channels, 16-bit resolution, which ensures that the current inputs of 30 coils can be regulated simultaneously. OPA549 is employed to convert the analog voltage signals from AD5370 to
the current input. The current feedback is obtained by measuring the voltage drop of the current sampling resistor. Then the A/D chip ADS8364 is used for the digitalization of the voltage signal. 2. Experiment on spinning motion control
Based on the control method described in Sec. IV A, the spinning control experiment is carried out. The initial orientation of the rotor is α = β = γ = 0, where the rotor frame and the stator frame are coincide. The initial position of the rotor in the spinning direction θ 0 is zero. The amplitude and frequency of the sine wave is set as A = 0.8, ωc = 16π . The current inputs for the middle layer coils are Isj = 0.8 sin(16π t − 0.8π (j − 1)),
1 ≤ j ≤ 5.
The spinning velocity is γ˙ = 4π (rad/s) (120 rpm). Fig. 9 shows the experiment results for this spinning motion control. It can be seen that the rotor following the desired trajectory well, which validate the effectiveness of the spinning control strategy. To check the performance of the spherical actuator under a low spinning velocity, extensive experiments are conducted.
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FIG. 13. Tilting control of spinning rotor with two layer coils. (a) Tilting motion. (b)Spinning motion. (c) Steady state tracking error of α and β.
The initial condition is the same as the previous experiment, the current patterns used to energize the coils are Isj = 0.8 · sin(π t − 0.8 · π · (j − 1)),
1 ≤ j ≤ 5.
The spinning velocity is γ˙ = π/4(rad/s) (7.5 rpm). The trajectory tracking result is shown in Fig. 10. We can see that the rotor have a good tracking performance, the motion is smooth. The results of experiment indicate that compared to the square wave used in literature,12 the designed sine wave control method has a good capability and can eliminate surging and oscillatory at low frequency. 3. Experiment on tilting motion control
To validate the tilting motion ability of the spherical actuator under the proposed control strategy, the step response experiment of titling motion control is implemented. The rotor is commanded from an initial orientation q0 = (0, 0, 0) to the position q1 = (4◦ , 8◦ , 0), and then the rotor is regulated to the orientation q2 = (8◦ , 11◦ , 0), q3 = (15◦ , 15◦ , 0), and q4 = (10◦ , 6◦ , 0) in turn. Finally, the rotor is controlled back to the upright orientation q0 = (0, 0, 0). The current inputs of the upper
and lower coils are calculated by Eq. (10). The controller gain matrices are Kp = diag[60, 60, 60], Kv = diag[12, 12, 12]. The experimental results are shown in Fig. 11. It shows that the tilting control without spinning motion has a highly accurate response and the rotor tracks the desired trajectory well. The steady state tracking error of the tilting motion is about 0.1◦ (Fig. 11(c)). To show the error curve more clearly, only a portion of the curve are given. The results also demonstrate the correctness of the dynamic and torque analysis. 4. Experiment on tilt control of spinning rotor
This section introduces the experiments on the tilt control of the spinning rotor. The trajectory is set as follows: first, the rotor is commanded to spinning at 60 rpm in the upright position (α = 0, β = 0), the parameters of the current patterns in Eq. (6) are A = 0.6, ωc = 8π . Then, the spinning rotor is driven to tilt to the orientation: o1 : (α = −5◦ , β = 5◦ ), o2 : (α = 5◦ , β = −5◦ ), o3 : (α = 15◦ , β = 0◦ ), and o4 : (α = 10◦ , β = 5◦ ) in turn. Finally, the rotor is regulated to back to the initial upright orientation. The current inputs used to energize the tilting motion are computed by Eq. (10). Dur-
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ing this process, the rotor is maintained spinning at 60 rpm around rotor shaft. To compare the performance under different control strategy, the experiment on the control with two layer coils is also conducted, where both the spinning and tilting motion are manipulated by the upper and lower layer coils. In this case, the current supplied to each coil consists of two parts: the tilting current and the spinning current. The current computation method of each part is the same as the control with three layer coils. The experiment results are given in Figs. 12 and 13. We can see that when the actuator is controlled with two layer coils, the steady state tracking error is about 0.8◦ (Fig. 13(c)). However, when it is controlled with three layer coils under the decoupling control strategy, the error is about 0.42◦ (Fig. 12(c)). The results demonstrate that the proposed decoupling control strategy can effectively improve the control accuracy of the actuator, and the coupling effect between the tilting and spinning motion can be reduced. The tracking error may be caused by the friction torque and the manufacturing errors exist in the mechanical structure. The rotor motion is also found to be nonconcentric, which will influence the control performance. The experimental results demonstrate the effectiveness of the design and the control strategy. They also provide insights for our future study. To overcome the mechanical errors exist in the system, a calibration algorithm should be investigated. In addition, the nonlinear friction torque should be evaluated and included in the control model for the high accuracy control.
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VI. CONCLUSIONS
In this paper, a spherical actuator which can produce three-DOF motion in one joint has been introduced. The dynamic and the torque characteristics of the actuator have been analyzed. The proposed control strategy in this paper can decouple the tilt control of rotor from its spinning motion, which allows us to control them independently. To reduce the coupling between the spinning and tilting motion, they are excited by different coils. Specifically, the spinning motion based on step control is governed by middle layer coils; while the tilting motion based on computed torque control law is regulated by upper and lower layer coils. The prototype and the experimental platform have been developed, and the validity of the analysis and the control strategy has been verified on it. ACKNOWLEDGMENTS
This work was supported by National Nature Science Foundation of China under Grant No. 50975017, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20101102110006 and Innovation Foundation of BUAA (Beijing University of Aeronautics and Astronautics) for Ph.D. Graduates. APPENDIX: CORIOLIS MATRIX OF THE DYNAMIC EQUATION
˙ in Eq. (3) are The elements of the Coriolis matrix C(q, q)
C11 = (J1 cβsβc2 γ − J2 cβsβs 2 γ + J3 sβcβ)β˙ + (−J1 sγ cγ c2 β + J2 c2 βsγ cγ )γ˙ , 1 C12 = (−J1 cβsβc2 γ − J2 cβsβs 2 γ + J3 sβcβ)α˙ − ((J1 − J2 )sβcγ sγ )β˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ + J3 sβ)γ˙ , 2 1 ˙ C13 = −((J1 − J2 )cγ sγ c2 β)α˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ + J3 cβ)β, 2 1 C21 = (J1 cβsβc2 γ + J2 cβsβs 2 γ − J3 sβcβ)α˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ − J3 cβ)γ˙ , 2 C22 = ((J1 − J2 )cγ sγ )γ˙ , 1 ˙ (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ − J3 cβ)α˙ + ((J1 − J2 )cγ sγ )β, 2 1 ˙ = ((J1 − J2 )cγ sγ c2 β)α˙ + ((J1 − J2 )cβs 2 γ − (J1 − J2 )cβc2 γ − J3 cβ)β, 2 1 ˙ = ((J1 − J2 )cβs 2 γ − (J1 − J2 )cβc2 γ + J3 cβ)α˙ − ((J1 − J2 )cγ sγ )β, 2 = 0.
C23 = C31 C32 C33
1 K.
M. Lee, H. S. Sun, and J. Joni, Proceedings of IEEE International Conference on Robotics and Automation, Barcelona, Spain, April 2005 (IEEE, 2005), p. 3652. 2 H. Y. Kim, H. C. Kim, and D. Gweon, Sens. Actuators, A 195, 38 (2013). 3 H. Nagasawa and S. Honda, Proceedings of American Society for Precision Engineering, Scottsdale, Arizon, 2000 (ASPE, 2000), p. 219. 4 L. Howald, H. Rudin, and H.-J. Güntherodt, Rev. Sci. Instrum. 63, 3909 (1992).
5 Y.
Zhang, W. J. Zhang, J. Hesselbach, and H. Kerle, Rev. Sci. Instrum. 77, 035112 (2006). 6 L. Yan, I. M. Chen, C. K. Lim, G. Yang, and K. M. Lee, Mechanisms and Machine Science (Springer, Netherlands, 2011), Vol. 4, p. 7. 7 F. C. Williams, E. R. Laithwaite, and J. F. Eastham, Proceedings of the Institution of Electrical Engineers,1959 (IEEE, 1959), p. 471. 8 L. Yan, I. M. Chen, C. K. Lim, G. Yang, and K. M. Lee, IEEE/ASME Trans. Mechatron. 17, 1080 (2012).
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Yan, I. M. Chen, C. K. Lim, G. Yang, W. Lin, and K. M. Lee, Mechatronics 21, 85 (2011). 10 K. M. Lee and H. Son, Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Monterey, California, July 2005 (IEEE, 2005), p. 335. 11 D. W. Kang, W. H. Kim, S. C. Go, C. S. Jin, S. H. Won, D. H. Koo, and J. Lee, IEEE Trans. Magn. 45, 2819 (2009). 12 H. Son and K. M. Lee, IEEE Trans. Ind. Electron. 57, 3475 (2010). 13 L. Rossini, O. Chételat, E. Onillon, and Y. Perriard, International Conference on Electrical Machines and Systems, Beijing, China, 2011 (IEEE, 2011), p. 1. 14 Y. Öner, Sens. Actuators, A 137, 200 (2007). 15 S. Maeda, K. Hirata, S. Ikejiri, and M. Tong, XIX International Conference on Electrical Machines, Rome, Italy (IEEE, 2010), p. 1.
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Kahlen, I. Voss, C. Priebe, and R. W. D. Doncker, IEEE Trans. Power Electron. 19, 1628 (2004). 17 W. Chen, L. Zhang, L. Yan, and J. Liu, Sens. Actuators, A 180, 75 (2012). 18 B. Li, G. D. Li, and H. F. Li, IEEE Trans. Magn. 47, 2127 (2011). 19 L. Yan, I. M. Chen, H. Son, C. K. Lim, and G. Yang, IEEE/ASME Trans. Mechatron. 15, 986 (2010). 20 J. Xiao, J. Xiao, N. Xi, H. Dulimartad, R. L. Tummalae, M. Minorf, and R. Mukherjeeg, Integr. Comput. Aided Eng. 11, 289 (2004). 21 R. M. Murray, Z. X. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, 1994). 22 L. Zhang, W. Chen, L. Yan, and J. Liu, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA September 2011 (IEEE, 2011), p. 744.
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125001 (2013)
Analysis and decoupling control of a permanent magnet spherical actuator Liang Zhang, Weihai Chen,a) Jingmeng Liu, and Xingming Wu School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
(Received 17 July 2013; accepted 12 November 2013; published online 5 December 2013) This paper presents the analysis and decoupling control of a spherical actuator, which is capable of performing three degree-of-freedom motion in one joint. The proposed actuator consists of a rotor with multiple PM (Permanent Magnet) poles in a circle and a stator with circumferential coils in three layers. Based on this actuator design, a decoupling control approach is developed. Unlike existing control methods that each coil is responsible for both the spinning and tilting motion, the proposed control strategy specifies the function of each coil. Specifically, the spinning motion is governed by the middle layer coils with a step control approach; while the tilting motion is regulated by upper and lower coils with a computed torque control method. Experiments have been conducted on the prototype to verify the validity of the design procedure, and the experimental results demonstrate the effectiveness of the analysis and control strategy. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4833681] I. INTRODUCTION
Many industrial applications such as manufacturing, helicopters or underwater vehicles, automobile wheel, and robotic joints require orientation control of the rotating shaft.1 Conventionally, this type motion is achieved by connecting a few single-axis actuators in series or in parallel with external mechanism. However, this combined actuation system has intrinsic disadvantages such as bulky structure, large backlash, slow dynamic response, singularity existence in workspace, and lack of dexterity.2 To overcome these drawbacks, the researchers have proposed the concept of spherical actuator, which can realize three-DOF (degree-of-freedom) motion in one joint. Currently, several different driving techniques such as the mechanical methods, piezoelectricity, and electromagnetic forces have been proposed for the spherical actuators.3–6 Electromagnetic actuation principle has been paid more attention among them.6 The earliest spherical induction motor is proposed by Williams et al.7 It can generate 2-DOF motion in its workspace. After that, especially in recent decades, various designs and control methods have been proposed.8–18 Chen et al. have made a spherical actuator consisting of a rotor with eight PM (Permanent Magnet) poles and a stator with 24 aircore coils. The magnetic field and torque model have been analyzed, but the control performance is not included.8, 9 Lee et al. have proposed a design concept of the spherical wheel motor,10 a similar design has been developed by Kang et al.11 Its open loop control method has been proposed, while the square wave used for the spinning motion results in torque ripple and deteriorates the control accuracy.12 Rossini et al. have presented the open-loop control of a reaction spherical actuator, which can be used for the satellite attitude control.13 Öner has proposed a four pole spherical rotor, and its magnetic field and open loop control method have been analyzed.14 Maeda a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2013/84(12)/125001/10/$30.00
et al. have developed a spherical actuator consists of a rotor with four interior magnets and two sets of stators with six magnetic poles. The feed-back control has been studied, while the control performance is affected by the accumulated orientation measurement error.15 Kahlen et al. have developed a torque control method for the spherical motor, which consists of a rotor with 112 PM poles and stator with 96 coils.16 In conclusion, one common feature of the existing spherical actuators is that many coils are mounted on the stator in 3D space.19 If a certain coil is used to generate 3-DOF rotational motion at the same time, the required current input changes of spinning motion will inevitably influence the motion in the tilting direction.12 Therefore, the control accuracy of the spherical actuator will be affected. In this paper, we will present an alternative control strategy for the spherical actuator. Unlike the existing energizing strategy that each coil governs all the 3-DOF rotations, this approach specifies the function of each coil. Specifically, the coils are evenly grouped into three layers, the middle layer coils are used for the spinning motion, and the upper and the lower layer coils are used for the tilting motion. By doing so, the control of the spherical actuator becomes easier, and the effect of the coupling between the tilting and spinning motion can be reduced. In addition, since one more layer coils are assembled in the inner space of the stator, the force density of this actuator is better than the spherical actuator with two layer coils. The control method presented in this paper can decouple the tilt control of the rotor from its spinning motion. The spinning motion is controlled with step control method, and sine wave is used to reduce the oscillation and the undesired torque ripple. The tilting motion is controlled with computed torque control method, which is a model-based dynamic control method. The reminder of the paper is organized as follows. First, the structure of the spherical actuator is described. Second we analyze the dynamic and torque characteristics. After that, the control strategies for the spinning and tilting motion are designed. Then we illustrate the prototype of the proposed spherical actuator and its experimental platform, on which the
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© 2013 AIP Publishing LLC
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Rotor shaft Stator
Rotor
Passive spherical joint
Base
(a)
(b)
FIG. 1. Spherical actuator. (a) CAD model of spherical actuator. (b) Assembly diagram.
experimental studies are carried out. Finally, the paper is summarized in conclusions. II. STRUCTURE DESCRIPTION
Compared to the conventional 1-DOF actuators, the design of a 3-DOF spherical actuator is quite different. Fig. 1(a) shows the CAD (Computer Aided Design) model of the proposed spherical actuator, which consists of a rotor housed in a spherical shell shaped stator. Fig. 1(b) shows the assembly diagram of the spherical actuator. A. Rotor
Fig. 2 shows the structure of the rotor. Eight PM poles are uniformly assembled on the equatorial plane of the rotor. The unit vector on the magnetization axis of PM poles in the rotor frame is given by ri = (−1)
i−1
[ cos ϕi
sin ϕi
T
0] ,
i = 1, 2, . . . , m, (1)
where ϕ are the azimuth angle of the PM pole and m is the number of PM poles. To reduce the inertial of the rotor, eight evenly distributed trapezoid holes are processed on the rotor shell. The rotor is supported and measured by the passive spherical joint (Fig. 3). This spherical joint is composed of a 1-DOF passive rotary joint in conjunction with a 2-DOF passive universal joint. An encoder and a 2-axes tilt sensor are
FIG. 2. Rotor structure.
FIG. 3. Passive spherical joint.
assembled on the passive spherical joint. The spinning angle of the rotor can be detected by the encoder, and the tilting angle can be measured by the 2-axes tilt sensor. With this measurement method, a closed-loop control can be implemented, and thus the controllable of the actuator can be effectively improved. B. Stator
Fig. 4 shows the exploded view of the stator. Three layer holes are distributed on the stator shell and 30 coils are mounted in these holes. The middle layer coils are evenly mounted on the equatorial plane of the stator, and the upper
FIG. 4. Stator structure.
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FIG. 5. Operating principle. (a) Spinning motion. (b) Tilting motion.
and lower coils are symmetrically assembled respect to the stator equatorial plane. Because of this design, the proposed decoupling control strategy can be implemented. Moreover, because one more layer coils are distributed on the stator, the inner space of the stator is better used. Thus, the torque output ability is improved. The unit vector on the coils axis in the stator frame is given by sj = [cos ηj cos φj j = 1, 2, . . . , n,
cos ηj sin φj
sin ηj ]T ,
(2)
where η and φ are the polar and azimuth angle of the coil axis, n is the number of coils. C. Operating principle
Spherical actuator can realize three-DOF motion in one single joint. Here, the spinning motion is controlled by the middle layer coils, and the tilting motion is controlled by the upper and lower coils. Current-carrying coils act like a current controlled magnet and establish a magnetic field around it. The force is generated as a result of the interaction between the stator magnetic field and the rotor magnetic field. The symbols “N” and “S” in Fig. 5 indicate the north and south polarities, respectively. The operation principle of the spinning motion is similar to the stepper motor. Ten middle layer coils are divided into five phases to control the spinning motion. The rotor has four north and four south pole permanent magnets. The polarity of the coils depends on the current input. If the coils are energized as shown in Fig. 5(a), an anticlockwise rotation along the rotor shaft can be generated. If the current inputs of these ten coils are changed in sequence as a stepper motor, a continuous spinning motion can be generated. A change in current input from one state to another creates a change in the rotor position. If the current is not change, the rotor stays in that stable position. The operating principle of the two-DOF tilting motion is shown in Fig. 5(b). The rotor will tilt around the axis normal to the plan under the tilting torque if the coils are energized as shown in Fig. 5(b). The current inputs are regulated according to the desired rotor orientation and tilting torque. Any changes of the current inputs will result in a differential
torque that drives the rotor to its equilibrium. By varying the current inputs based on the control model, the desired tilting motion can be realized. If we energize all the three layer coils at the same time, three-DOF rotational motion can be generated. The torque model of the spherical actuator is orientation dependent, and tilting control is based on the torque model. Thus, when we control the spinning and tilting motion at the same time, the orientation changes from the spinning motion should be included in the tilting control algorithm. In our control model, the torque computation is involved in the tilting control law, and the orientation information can be obtained by the passive spherical joint. Therefore, by regulate the current inputs of all the coils under the control law, the desirable three-DOF rotational motion in the workspace can be achieved. III. DYNAMIC AND TORQUE ANALYSIS A. Rotor dynamics
XYZ Euler angles (α, β, γ ) are used to express the rotor orientation, which has no singularities in the workspace of the actuator.20 xr yr zr and x0 y0 z0 are coordinates on the rotor and stator, respectively. The zr -axis of the rotor is pointing along the rotor shaft. The rotor is a single rigid body mechanical system. Its dynamical equation derived from the Lagrange’s equation in terms of Euler angle (α, β, γ ) is given by ˙ q˙ = τ c − τ ext , M(q)q¨ + C(q, q) ⎡ J1 c2 βc2 γ +J2 c2 βs 2 γ +J3 s 2 β
where M(q) = ⎣
(J1 −J2 )cβcγ sγ J3 sβ
(J1 −J2 )cβcγ sγ
J1 s 2 γ +J2 c2 γ
⎡
0
(3) ⎤
J3 sβ 0
⎤
⎦
J3
C11 C12 C13 ˙ = ⎣ C21 C22 C23 ⎦ is the is the inertial matrix; C(q, q) C31 C32 C33 Coriolis matrix; q = [ α β γ ]T is the Euler angle vector; T τ c = [τα τβ τγ ] is the control torque; τ ext is the external torque including the load and disturbances; and J1 = Jxx , J2 = Jyy , J3 = Jzz are the principle inertial moments of the ro˙ are given tor. The elements of the Coriolis matrix C(q, q) in the Appendix. It can be seen from Eq. (3) that the rotor
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dynamics has nonlinear characteristic, many interaxis coupling terms exist in the dynamic model.
B. Torque calculation
The torque here is analyzed using the finite-element (FE) method. The FE model is established in Maxwell 3D. Since the coils are air-core, the torque has a linear property with respect to the current input, and the torque generated by all the coils can be calculated by summing up each individual coils.9 The total torque for all the coils of the actuator is T = [ Tx Ty Tz ]T = GI,
(4)
where the torque matrix G = [ G1 . . . Gj . . . GN ], Gj ∈ 3×1 and the current input I = [ I1 . . . Ij . . . In ]T . Gj is the torque coefficient, which describes the torque contribution of the j th coil. n is the number of the coils, and Ij is the current input of the jth coil. Gj is given as ⎧ m ⎨ (−1)i−1 f (ϕ )(r × s /r × s ), if r × s = 0 ij i j i j i j Gj = i=1 ⎩ 0, if ri × sj = 0, (5) where f(ϕ ij ) is the torque constant function derived by curve fitting the torque between one PM pole and one coil, ϕ ij = cos −1 (ri · sj ) is the separation angle between the PM pole and the coil. It can be seen from Eqs. (4) and (5) that the torque output of the spherical actuator is orientation dependent. IV. CONTROL DESIGN OF THE SPHERICAL ACTUATOR
A decoupling control method which decouples the tilt control of the rotor from its spinning motion is proposed. The spinning motion is controlled with step control method by the middle layer coils, and the tilting motion is controlled with computed torque control law by the upper and lower lay coils. Since that if one coil is used to regulate both the spinning and tilting motion at the same time, the demanded current changes in spinning motion will inevitably affect the motion in tilt direction. Thus, the function of each coil is specified. Such design can not only makes the control easier but also reduce the coupling between the motions in different direction.
A. Spinning motion control
The spinning motion control here is similar to the stepper motor, a given coil excitation state defines a rotor position as a result of the electromagnetic force between the stator coils and the PM poles. The diametrically opposite coils are arranged in one phase and connected in series. Ten middle layer coils are divided into five phases as Phase 1 : C1 , C6 ;
Phase 2 : C2 , C7 ;
Phase 4 : C4 , C9 ;
Phase 5 : C5 , C10 ;
Phase 3 : C3 , C8 ;
where Cj indicates the jth coil (each middle layer coil is given a label as shown in Fig. 5). The minimum step size is 9◦ , which is determined by the number of PM poles and middle layer coils. To reduce the oscillation and the undesired torque ripple, sine wave is used for the spinning motion instead of square wave. The sine wave current input does not change in sudden jumps, but instead it is smoothly changed. This will bring advantages as smoother motion and electronically controlled finer step sizes. In addition, the step-loss problem can be reduced. The current input of the jth phase is given as
4 (6) Isj = A sin ωc t + θ0 − π (j − 1) , 1 ≤ j ≤ 5, 5 where A and ωc are the amplitude and frequency of the sine wave, θ 0 indicates the initial position of the rotor in the spinning direction. The spinning period of the spherical actuator can be derived from the current input, which is given as ts = 8π /ωc . Thus, by controlling the frequency of the sine wave, the spinning rate can be regulated. The step loss can be detected by the encoder and will be compensated when necessary. B. Tilting motion control
Tilting motion is a distinct feature of the spherical actuator. The control task here is to determine the required current inputs that cause the rotor to execute a desired tilting motion. The question is divided into two parts: first, determining the applied driving torque that drives the rotor to track a desired tilting motion; then, the determination of the optimal current inputs that generates the desired driving torque. 1. Computed torque control for tilting motion
The objective of the dynamic control algorithm is to determine the driving torque that actuates the rotor to follow the commanded motion. Considering the dynamic characteristic of the spherical actuator, computed torque control method is utilized for the tilting control of the spherical actuator.21 The desired driving toque Tt = [ Tt1 Tt2 0 ]T which actuates the rotor from its initial equilibrium state to the final state under the computed torque control method is given by ˙ q, ˙ Tt = M(q)(q¨ d − Kv e˙ − Kp e) + C(q, q)
(7)
where q = [ α β 0 ] express the tilting orientation of the rotor and qd is the desired tilting orientation; e(t) = q(t) ˙ − q˙ d (t) define the tracking error and − qd (t) and e˙ (t) = q(t) its derivative, respectively; Kv and Kp are positive definite differential and proportional coefficient matrices, respectively. When substituting Eq. (7) into Eq. (3), the error dynamics can be written as T
M(q)(¨e + Kv e˙ + Kp e) = 0.
(8)
Since M(q) is positive definite, we have e¨ + Kv e˙ + Kp e = 0.
(9)
Equation (9) is a linear differential equation, which regulates the error between the desired and actual trajectories. Since
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FIG. 6. Prototype of the spherical actuator. (a) Spherical actuator. (b) Rotor. (c) Passive spherical joint.
Eq. (9) is linear, the positive definite gain matrices Kv and Kp are easy to choose to ensure the stability of the system. In addition, it can be deduced that e → 0 exponentially as t → ∞. 2. Current inputs control
The spherical actuator has more independent coils than the dimensional of the tilting torque. The redundant in the current inputs implies that an optimal control can be conducted. Here, the optimal current inputs of the upper and lower coils are obtained by minimizing the energy consumption, which is given by It = GT (GGT )−1 Tt .
(10)
If the calculated current inputs are smaller than the current limits, they can be used to energize the upper and lower coils directly, and the power consumption is minimized. However, when the calculated current inputs exceed the current limits, a fault tolerant control should be implemented to ensure the stability of the system.22 V. PROTOTYPE AND EXPERIMENTS A. Prototype
Fig. 6 shows the prototype of the proposed spherical actuator. The structure specifications are given in Table I. The material of the rotor and stator shell is aluminum, and the nonmetal material can be used in future to eliminate the eddy current effect. The working range of the tilting motion is ±15◦ , which can be increased by distributing more layer coils on the
FIG. 7. Experimental platform.
stator. The type of the encoder is Angtron-RE-20-485, which is a 14 bit absolute sensor. The type of 2-axes tilting sensor is SANG 1000, whose accuracy is up to 0.01◦ . The power consumption of the spherical actuator is about 135 W. Two cylindrical holes separated 90◦ apart that point to the stator center are designed at the equator plane of the stator shell. Similarly, two holes separated 90◦ apart are drilled at the equator plane of the rotor shell. Two centering pins can be inserted to these holes to ensure the concentricity between rotor and stator. B. Experiments
1. Experiment setup
To demonstrate the performance of the spherical actuator, an experimental platform consists of the research prototype and its control system is developed as shown in Fig. 7. The whole control system comprises two parts: the host computer and the current controller, they communicate with each other through the serial port. The host computer is mainly responsible for the control algorithm computation and communication with the current controller. A graphical user interface (GUI) program written with VC++ is developed on the host computer in the VS2010 environment, through which the control command can be sent to the host computer. In addition, the system information can
TABLE I. Structure specifications of the spherical actuator. Rotor Number of PM poles Inner/outer rotor shell Cylindrical PM pole Magnetization Moment of rotor inertia (kg/m2 × 10−3 ) Stator Number of coils Inner/outer stator shell Cylindrical coil Winding parameters
8 PM poles in one layers 53/58.5 mm Radius = 12.5 mm, height = 20 mm Axial magnetization, Br = 1.21 tesla J1 = 2.219, J2 = 2.176, J3 = 2.256
30 coils in three layer 86/98 mm Radius = 11 mm, height = 25 mm AWG 26 copper wire, 950 turns FIG. 8. Diagram of current controller.
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FIG. 9. Spinning motion tracking.
Rev. Sci. Instrum. 84, 125001 (2013)
FIG. 10. Spinning motion at low velocity.
FIG. 11. Step response of tilting control. (a) Euler angle α. (b) Euler angle β. (c) Steady state tracking error of α and β.
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FIG. 12. Tilting control of spinning rotor with three layer coils. (a) Tilting motion. (b) Spinning motion. (c) Steady state tracking error of α and β.
be displayed in real time on it, such as the orientation, current, temperature, spinning velocity, etc. According to the control command, the host computer does the algorithm calculation and sends the corresponding command to the current controller through the serial port. The current controller is in charge of providing multichannel, bipolar current inputs to the stator coils; sampling and processing the orientation and current information; sending the information to the host computer for motion control and real-time display. As shown in Fig. 8, the current controller system consists of two parts: the core control part and the power amplifier output module. The core control part is comprised of ARM (Advanced RISC Machines) and CPLD (Complex Programmable Logic Device). The ARM7 microprocessor is mainly in charge of task scheduling, and the CPLD is mainly responsible for driver function programming. The power amplifier output module is responsible for providing the current to the coils. Since the input current should be multi-channel and bipolar, the D/A chip AD5370 is utilized here. It could offer digital-to-analog converting with 40 channels, 16-bit resolution, which ensures that the current inputs of 30 coils can be regulated simultaneously. OPA549 is employed to convert the analog voltage signals from AD5370 to
the current input. The current feedback is obtained by measuring the voltage drop of the current sampling resistor. Then the A/D chip ADS8364 is used for the digitalization of the voltage signal. 2. Experiment on spinning motion control
Based on the control method described in Sec. IV A, the spinning control experiment is carried out. The initial orientation of the rotor is α = β = γ = 0, where the rotor frame and the stator frame are coincide. The initial position of the rotor in the spinning direction θ 0 is zero. The amplitude and frequency of the sine wave is set as A = 0.8, ωc = 16π . The current inputs for the middle layer coils are Isj = 0.8 sin(16π t − 0.8π (j − 1)),
1 ≤ j ≤ 5.
The spinning velocity is γ˙ = 4π (rad/s) (120 rpm). Fig. 9 shows the experiment results for this spinning motion control. It can be seen that the rotor following the desired trajectory well, which validate the effectiveness of the spinning control strategy. To check the performance of the spherical actuator under a low spinning velocity, extensive experiments are conducted.
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FIG. 13. Tilting control of spinning rotor with two layer coils. (a) Tilting motion. (b)Spinning motion. (c) Steady state tracking error of α and β.
The initial condition is the same as the previous experiment, the current patterns used to energize the coils are Isj = 0.8 · sin(π t − 0.8 · π · (j − 1)),
1 ≤ j ≤ 5.
The spinning velocity is γ˙ = π/4(rad/s) (7.5 rpm). The trajectory tracking result is shown in Fig. 10. We can see that the rotor have a good tracking performance, the motion is smooth. The results of experiment indicate that compared to the square wave used in literature,12 the designed sine wave control method has a good capability and can eliminate surging and oscillatory at low frequency. 3. Experiment on tilting motion control
To validate the tilting motion ability of the spherical actuator under the proposed control strategy, the step response experiment of titling motion control is implemented. The rotor is commanded from an initial orientation q0 = (0, 0, 0) to the position q1 = (4◦ , 8◦ , 0), and then the rotor is regulated to the orientation q2 = (8◦ , 11◦ , 0), q3 = (15◦ , 15◦ , 0), and q4 = (10◦ , 6◦ , 0) in turn. Finally, the rotor is controlled back to the upright orientation q0 = (0, 0, 0). The current inputs of the upper
and lower coils are calculated by Eq. (10). The controller gain matrices are Kp = diag[60, 60, 60], Kv = diag[12, 12, 12]. The experimental results are shown in Fig. 11. It shows that the tilting control without spinning motion has a highly accurate response and the rotor tracks the desired trajectory well. The steady state tracking error of the tilting motion is about 0.1◦ (Fig. 11(c)). To show the error curve more clearly, only a portion of the curve are given. The results also demonstrate the correctness of the dynamic and torque analysis. 4. Experiment on tilt control of spinning rotor
This section introduces the experiments on the tilt control of the spinning rotor. The trajectory is set as follows: first, the rotor is commanded to spinning at 60 rpm in the upright position (α = 0, β = 0), the parameters of the current patterns in Eq. (6) are A = 0.6, ωc = 8π . Then, the spinning rotor is driven to tilt to the orientation: o1 : (α = −5◦ , β = 5◦ ), o2 : (α = 5◦ , β = −5◦ ), o3 : (α = 15◦ , β = 0◦ ), and o4 : (α = 10◦ , β = 5◦ ) in turn. Finally, the rotor is regulated to back to the initial upright orientation. The current inputs used to energize the tilting motion are computed by Eq. (10). Dur-
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ing this process, the rotor is maintained spinning at 60 rpm around rotor shaft. To compare the performance under different control strategy, the experiment on the control with two layer coils is also conducted, where both the spinning and tilting motion are manipulated by the upper and lower layer coils. In this case, the current supplied to each coil consists of two parts: the tilting current and the spinning current. The current computation method of each part is the same as the control with three layer coils. The experiment results are given in Figs. 12 and 13. We can see that when the actuator is controlled with two layer coils, the steady state tracking error is about 0.8◦ (Fig. 13(c)). However, when it is controlled with three layer coils under the decoupling control strategy, the error is about 0.42◦ (Fig. 12(c)). The results demonstrate that the proposed decoupling control strategy can effectively improve the control accuracy of the actuator, and the coupling effect between the tilting and spinning motion can be reduced. The tracking error may be caused by the friction torque and the manufacturing errors exist in the mechanical structure. The rotor motion is also found to be nonconcentric, which will influence the control performance. The experimental results demonstrate the effectiveness of the design and the control strategy. They also provide insights for our future study. To overcome the mechanical errors exist in the system, a calibration algorithm should be investigated. In addition, the nonlinear friction torque should be evaluated and included in the control model for the high accuracy control.
Rev. Sci. Instrum. 84, 125001 (2013)
VI. CONCLUSIONS
In this paper, a spherical actuator which can produce three-DOF motion in one joint has been introduced. The dynamic and the torque characteristics of the actuator have been analyzed. The proposed control strategy in this paper can decouple the tilt control of rotor from its spinning motion, which allows us to control them independently. To reduce the coupling between the spinning and tilting motion, they are excited by different coils. Specifically, the spinning motion based on step control is governed by middle layer coils; while the tilting motion based on computed torque control law is regulated by upper and lower layer coils. The prototype and the experimental platform have been developed, and the validity of the analysis and the control strategy has been verified on it. ACKNOWLEDGMENTS
This work was supported by National Nature Science Foundation of China under Grant No. 50975017, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20101102110006 and Innovation Foundation of BUAA (Beijing University of Aeronautics and Astronautics) for Ph.D. Graduates. APPENDIX: CORIOLIS MATRIX OF THE DYNAMIC EQUATION
˙ in Eq. (3) are The elements of the Coriolis matrix C(q, q)
C11 = (J1 cβsβc2 γ − J2 cβsβs 2 γ + J3 sβcβ)β˙ + (−J1 sγ cγ c2 β + J2 c2 βsγ cγ )γ˙ , 1 C12 = (−J1 cβsβc2 γ − J2 cβsβs 2 γ + J3 sβcβ)α˙ − ((J1 − J2 )sβcγ sγ )β˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ + J3 sβ)γ˙ , 2 1 ˙ C13 = −((J1 − J2 )cγ sγ c2 β)α˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ + J3 cβ)β, 2 1 C21 = (J1 cβsβc2 γ + J2 cβsβs 2 γ − J3 sβcβ)α˙ + (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ − J3 cβ)γ˙ , 2 C22 = ((J1 − J2 )cγ sγ )γ˙ , 1 ˙ (−(J1 − J2 )cβs 2 γ + (J1 − J2 )cβc2 γ − J3 cβ)α˙ + ((J1 − J2 )cγ sγ )β, 2 1 ˙ = ((J1 − J2 )cγ sγ c2 β)α˙ + ((J1 − J2 )cβs 2 γ − (J1 − J2 )cβc2 γ − J3 cβ)β, 2 1 ˙ = ((J1 − J2 )cβs 2 γ − (J1 − J2 )cβc2 γ + J3 cβ)α˙ − ((J1 − J2 )cγ sγ )β, 2 = 0.
C23 = C31 C32 C33
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