IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 6, DECEMBER 2011 1129
An Accelerometer Configuration for Reference-Frame-Independent Linear-State Estimation
Brandon G. Folk and Eric T. Wolbrecht, Member, IEEE
Abstract—This paper presents a novel three single-axis ac-celerometer configuration for measuring relative acceleration ina moving reference frame without knowledge of motion or orien-tation of the moving frame itself. Also presented is an extendedKalman filter (EKF) that combines this relative acceleration mea-surement with a linear-position measurement for state estimation.The motivation for this approach is the need for high-quality linear-state estimation in pneumatic cylinder control, but potential ap-plications of this approach are much broader, including mobilerobots and general relative linear-state sensing. When applied to apneumatic cylinder, this method eliminates the need for kinematicknowledge of the cylinder base and allows state estimation to beimplemented at the cylinder level without regard to the externalmotion of the robot. Experimental tests were performed to com-pare the presented reference-frame-independent EKF method to astandard kinematically dependent end-effector EKF. When kine-matic knowledge of the end effector is known, both standard andpresented methods perform well as expected. However, removingkinematic knowledge of the local reference frame adversely af-fects the performance of a standard kinematically dependent EKF,but does not affect the performance of the presented method as itdoes not depend on such global kinematic knowledge. Experimen-tal results also show that the addition of a direct relative velocitymeasurement does not significantly improve performance over thepresented method.
Index Terms—Accelerometer, extended Kalman filter (EKF),pneumatic cylinder, state estimation.
I. INTRODUCTION
ACCELERATION and velocity signals are becoming in-creasingly important as state feedback for nonlinear- and
high-bandwidth control systems, as noted by [1] and others.Such systems require accurate velocity and acceleration sig-nals that have high signal-to-noise ratios and low-to-zero phaselag. This requirement is difficult to obtain from the traditionalmethod of estimating velocity and acceleration by using numer-ical differentiation of a measured position signal. This approachamplifies signal noise, as noted by [2] and others. Some of thisnoise can be reduced by increasing the sampling rate, althoughat a significant increase in expense. A more common approachis to remove the noise created from numerical differentiation by
Manuscript received February 8, 2010; revised June 14, 2010; acceptedAugust 16, 2010. Date of publication October 21, 2010; date of current versionSeptember 7, 2011. Recommended by Technical Editor D. Sun.
The authors are with the Mechanical Engineering Department, Universityof Idaho, Moscow, ID 83843 USA (e-mail: [email protected];[email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2010.2080281
passing the measured position signal through a low-pass filter,which adds phase lag to the estimated velocity and accelerationsignals. In control applications, this alters the system dynamicsand reduces the controllable bandwidth of system, as detailedin [3].
A well-established approach to address the problems withnumerical differentiation is to measure acceleration directlyin addition to position and to fuse the measurements usingKalman filtering or other state-estimation techniques, as shownin [1], [4]–[15]. This approach has become a viable option dueto the availability of small, low-cost microelectromechanicalsystems accelerometers, and has been utilized in a broad rangeof applications. In [4], for example, an accelerometer is used asa driving function in an extended Kalman filter (EKF) to im-prove state estimation of a low-resolution optical encoder. In [5]and [6], onboard inertial sensors are used to avoid direct integra-tion of global positioning system signals for autonomous landvehicles. Furthermore in [7], low-cost accelerometers are usedto provide high-bandwidth tilt estimation. In a different applica-tion, tip velocity of a flexible-link robot is estimated by combingan endpoint accelerometer and a tip displacement sensor usingKalman filtering [8]. In an example of non-Kalman state esti-mation, Mahajan et al. [9] uses a fuzzy inference system to fusedata from multiple sensors, including an accelerometer.
For simple single degree-of-freedom (DOF) systems, stateestimation using acceleration and position signals is relativelystraightforward. For more complicated mechatronic and roboticsystems, the reference frame of the accelerometer must beknown in order to relate it to position measurements. A com-mon approach is to place a multiaxis accelerometer on the endeffector of the manipulator and to calculate the configurationand position of the accelerometer using the forward kinematicsof the manipulator, as in [15]. This technique typically requiresthe base of the manipulator to remain fixed with respect to theearth’s reference frame.
For two notable applications, however, this technique is im-possible and/or undesirable. The first, more general applicationis mobile robot systems (wheeled or legged). In such applica-tions, the exact configuration and position of the mobile robotmay not be directly known. Such a mobile robot may have amanipulator or robot arm for which accurate estimates of rel-ative angular or linear velocity and acceleration are important.Without complete forward kinematics, however, it is impossibleto relate the position and configuration of the manipulator’s endeffector to the measured acceleration.
The second application is preengineered mechatronic sub-systems. In such applications, such as sensor or actuator de-velopment, it is necessary to measure acceleration information
1130 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 6, DECEMBER 2011
relative to the application’s reference frame because the final useof the developed subsystem is unknown. Here, the advantage ofmeasuring relative acceleration is that relative state estimationmay be developed prior to the intended use of the estimated state.For example, consider directly measuring relative accelerationof a linear-position sensor (i.e., linear encoder or linear poten-tiometer). Using state-estimation techniques will produce a fullrelative linear-state estimate. The resulting linear-state sensorcould fully estimate relative linear position and velocity, andcould be used in any application without regard to the motionof the sensor itself. A sensor of this type could be used for stateestimation of each DOF of a multi-DOF manipulator. Then, us-ing the forward kinematics, the joint space-state estimates forposition and velocity can be mapped to the spatial frame of themanipulator.
One such preengineered actuator application is pneumaticcylinder control, which was the motivation for this research.The importance of accurate position, velocity, and accelerationsignals for pneumatic control approaches is well documented(for examples, see [16]–[19]). Previous work has shown thatmeasuring acceleration in pneumatic systems can improve per-formance [15], but this approach requires kinematic knowledgeof the manipulator, which is different for each application. Analternative approach is to measure the relative acceleration of thecylinder rod with respect to the cylinder body, so that it is pos-sible to determine the cylinder stroke acceleration regardless ofthe orientation and motion of the cylinder body. When combinedwith a linear-stroke sensor and implementing state-estimationtechniques, this approach provides accurate, low noise, and low-to-zero phase lag state signals for use in a pneumatic cylindercontroller using only information obtained in the local cylinderreference frame. This separates the pneumatic controller fromits potential application, so that position and force controllersmay be developed prior to use.
This paper focuses on the problem of directly measuringrelative linear acceleration, and using this measurement in com-bination with a relative linear-position measurement for stateestimation. A novel three single-axis accelerometer configura-tion for directly measuring the linear acceleration of a point withrespect to a moving reference frame is presented first.Second,an extended Kalman filter (EKF) is presented that combinesthis local relative acceleration measurement with a local posi-tion measurement for state estimation. This EKF includes anadaptive state for learning a combined accelerometer drift off-set. This approach does not require kinematic knowledge of themoving reference frame.
The motivation of this paper is to improve pneumatic positionand force control by providing more accurate position, velocity,and acceleration signals (with both reduced phase lag and noise).To this end, a 2-DOF pneumatic cylinder robotic test bed wascreated. This test bed was used to compare the developed EKFto the two traditional techniques for estimating linear states: 1)numerical differentiation and 2) kinematically dependent stateestimation.
In the following sections, an analysis of the accelerometerconfiguration is given, followed by the development of the EKF.Then, experimental results are presented that compare the de-
Fig. 1. Location of the accelerometers on the pneumatic cylinder. Two ac-celerometers are mounted on the body (located at A and B) and the one on therod (located at C). Vectors a, b, and c are located with respect to a fixed inertialframe (earth), d is the fixed distance between accelerometers located at A and B,ω is the absolute angular velocity of the cylinder, and s is the distance betweenaccelerometers B and C, which varies with the stroke length of the rod.
veloped EKF to more traditional approaches. Finally, results arediscussed and conclusions are presented.
II. THREE SINGLE-AXIS ACCELEROMETER CONFIGURATION
By properly configuring three single-axis accelerometers, itis possible to directly measure the relative linear accelerationof a point with respect to a moving reference frame. In thisconfiguration, the three accelerometers are placed along the lineof motion, with the first two fixed with respect to the movingframe and the third fixed to the moving point.
For the pneumatic cylinder application, which was the motiva-tion for this research, the developed accelerometer configurationincludes two accelerometers placed on the cylinder body, andone placed on the piston rod (see Fig. 1 in the following). Withthis configuration, it is possible to determine the acceleration ofthe rod s with respect to the body regardless of the orientationand acceleration of the cylinder body. The two accelerometerson the body, located at A and B, and the one on the rod, locatedat C, are mounted along the axis of the cylinder rod and withthe same local coordinate frame xyz. Also, the radial distance(from the accelerometer to the stroke axis) is the same for eachaccelerometer. The vectors a, b, and c are located with respectto a fixed inertial frame (earth), d is the fixed distance betweenaccelerometers located at A and B, and s is the distance betweenaccelerometers located at B and C, which varies with the strokelength of the rod.
The analysis of this configuration starts by defining the ac-celeration of a particle P in a moving coordinate system xyz.Following [20], the location vector r of the point P with respectto the inertial reference frame xyz is defined as follows:
r = q + ρ (1)
where q is the location of the moving reference frame withrespect to the inertial reference frame and ρ is the location ofthe moving point P with respect to the moving reference framexyz.
The acceleration of the point r is found according to
FOLK AND WOLBRECHT: ACCELEROMETER CONFIGURATION FOR REFERENCE-FRAME-INDEPENDENT LINEAR-STATE ESTIMATION 1131
where r, q, and ρ are as previously defined, ω is the ab-solute angular velocity of the xyz system, q is the abso-lute acceleration of the xyz system, and (ρ)r and (ρ)r arethe relative velocity and acceleration, respectively, of pointP , as viewed from the xyz system. Then, (2) is applied tothe configuration of Fig. 1 with q = b and r = a. With thisconfiguration, ω = [ωx ωy ωz ]T , ρ = [−d 0 0 ]T , and(ρ)r = (ρ)r = 0 because the distance d between accelerome-ters located at A and B is constant. The resulting equation is asfollows:
a − b =
⎡⎢⎣
d(ω2
y + ω2z
)
−d (ωz + ωxωy )
d (ωy − ωxωz )
⎤⎥⎦ . (3)
Similarly, (2) can be applied to the configuration of Fig. 1again except with r = c. In this case, the distance between theaccelerometers changes with the cylinder stroke, so that nowρ = [ s 0 0 ]T , and (ρ)r = [ s 0 0 ]T , where s and s arethe velocity and acceleration, respectively, of the cylinder rodwith respect to the cylinder body. The resulting equation is asfollows:
c − b =
⎡⎢⎣
s − s(ω2
y + ω2z
)
s (ωxωy + ωz ) + 2ωz s
s (ωxωz − ωy ) − 2ωy s
⎤⎥⎦ . (4)
Since the only direction of concern is along the stroke axis(the direction that the rod moves), the equations along the localx-direction are selected from (3) and (4), so that
ax − bx = d(ω2
y + ω2z
)
cx − bx = s − s(ω2
y + ω2z
). (5)
Combining the two equations in (5) and solving for s givesthe acceleration of the cylinder rod with respect to the cylinderbody as a function of the three accelerations along the cylinderaxis measured in the local cylinder frames and the stroke lengthmeasurement
s = cx − bx +s
d
(ax − bx
). (6)
The result in (6) shows the arrangement of the three single-axis accelerometers, as shown in Fig. 1, allows the accelerationof the rod with respect to the cylinder body to be determinedregardless of the motion and/or orientation of the cylinder body.
III. EKF FOR THE THREE ACCELEROMETER CONFIGURATION
A Kalman filter uses a linear-system model and the measuredvariables to predict the states of the system. A Kalman filter canalso learn an offset or bias of an accelerometer placed at the endeffector, as described in [15], depending on the observabilityof the system [21]. Because of the nonlinearity of the threesingle-axis accelerometer configuration, it is necessary to usethe EKF, which linearizes the system about the current state.The following sections describe the EKF developed for the threesingle-axis accelerometer configuration.
A. EKF With Two Accelerometer Offsets
After calibrating the accelerometers, a bias, or offset mayexist due to the presence of drift in the sensor [22]. Therefore,we define actual accelerations ax = acca + δa , bx = accb + δb ,and cx = accc + δc , where δa , δb , and δc are the respectiveoffsets, and acca , accb , and accc are the respective accelerationmeasurements. With these definitions, (6) becomes
accab = acca − accb , and acccb = accc − accb , (7) can be sim-plified to give
s =[δcb +
s
dδab
]+ acccb +
s
daccab . (8)
The presence of the nonlinear s/d term in (8) requires theuse of an EKF for the state estimation. Along with position andvelocity states, the two combined offset terms are included inthe system such that the states are defined as follows:
x = [ δab δcb s s ]T . (9)
To learn the two combined accelerometer offsets, the systemmust be observable. For a linear system, observability is deter-mined from the system state and output matrices. For a nonlin-ear system, a sufficient condition for local observability can befound by performing the same analysis with the local linearizedsystem state and output matrices [23]. Using this analysis, thetwo offset system is not locally observable, but being a sufficientcondition, it was necessary to confirm the unobservability ex-perimentally. When extensive experiments showed the systemto be unobservable, it was necessary to design an EKF with onlyone accelerometer offset. The derivation of this EKF is given inthe following section.
B. Observable EKF via Grouped Accelerometer Offset
The unobservability of two offset EKF made it necessary tocombine the acceleration offsets into a single term, such that
δ ∼= δcb +s
dδab . (10)
This approximation allows the EKF to learn only one offsetδ, which produces an observable system, capable of adaptingto drift in the accelerometer offsets. This is an approximation,because when s changes so should δ. However, in practice,this EKF is able to change δ as necessary to achieve goodperformance.
It is now possible to substitute (10) into (8) giving an accel-eration equation with only one offset
s = δ + acccb +s
daccab . (11)
Since there is only one offset, the system now has the statesas follows:
x = [ δ s s ]T (12)
where δ, s, and s are as previously defined. Following the meth-ods described in [21], the system at the current time step k is
1132 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 6, DECEMBER 2011
defined as follows:
xk = f(xk−1 ,uk ,wk−1)
yk = h(xk ,vk ) (13)
where xk are the system states from (12), uk are the drivingfunctions (accelerometer signals in this application), wk is pro-cess noise, and vk is measurement noise. The discrete systemdynamics f for this application are as follows:
xk =
⎡⎢⎣
δ
s + ts (s)
s + ts(δ + (acccb + wcb) + s
d (accab + wab))
⎤⎥⎦
k−1(14)
where ts is the time step, and the other variables are as previouslydefined. The system output h is defined as follows:
yk = [s + vs ]k−1 . (15)
In this EKF design, the accelerometer signals in (14) areused to predict the state at the next time step, and then, thesystem output in (15), which is the signal from the linear-motionpotentiometer (LMP), is used to correct the prediction. Thisprediction and correction process, as described in [21], requiresdefining a process noise covariance matrixQ and a measurementnoise covariance matrix R. In some cases, it is possible todirectly measure these matrices. However, in many applications,these values are difficult to find; therefore, in practice, the Qand R matrices may be interpreted as gains of the EKF thatcontrol the balance between how the EKF relies on the drivingfunctions uk , and the measurements yk . In general, large Qvalues indicate large uncertainty in the process model and largeR values indicate large uncertainty in the measurements. In thisapplication, these matrices are tuned with the goal of a producingan accurate, low-noise velocity signal from the EKF. For thisapplication, a 30-Hz second-order low-pass Butterworth filterwas chosen as a benchmark for allowable signal noise. Thevalues for Q and R were chosen via experimental tuning toachieve good performance, and are given as follows
Q =
⎡⎢⎣
10000 1000 1000
1000 100000 1000
1000 1000 100000
⎤⎥⎦
R = 0.001. (16)
IV. EXPERIMENTAL RESULTS
Experiments were conducted to test the three single-axisaccelerometers configuration and corresponding EKF (3ACCEKF). In the experiments, the 3ACC EKF was compared tofour other methods for estimating relative linear states; 1) stateestimation using numerical derivatives of a position measure-ment; 2) state estimation using a kinematically dependent EKFwith accelerometers placed on the end effector; 3) direct localrelative velocity measurement; and 4) adding the direct velocitymeasurement to the presented 3ACC EKF. Descriptions of each
TABLE IDESCRIPTION OF THE PRESENTED NOVEL LOCAL STATE-ESTIMATION
METHOD (3ACC EKF) AND ALTERNATE METHODS USED FOR
EXPERIMENTAL COMPARISON
method are given in the following Table I and the abbreviatedlabels will be used in future sections to refer to each method.
The presented and alternate state-estimation methods listed inthe table earlier were tested using a 2-DOF manipulator, whichis described in Section IV-A. These experiments were intendedto demonstrate the following characteristics of the presentedtechnique (3ACC EKF).
1) When kinematic knowledge of the manipulator (i.e.,global orientation and motion of the accelerometers) isknown, the performance of the 3ACC EKF is comparableto that of end-effector EKF (EE EKF).
2) When kinematic knowledge of the manipulator (i.e.,global orientation and motion of the accelerometers) isremoved, the EE EKF becomes inaccurate, while the per-formance of the 3ACC EKF is unaffected.
3) The 3ACC EKF produces a velocity estimate that is asaccurate as measuring velocity directly.
4) The addition of the velocity measurement to the 3ACCEKF does not significantly improve estimation of positionand velocity.
The results from these experiments are presented inSection IV-B and IV-C, following the presentation of the ex-perimental apparatus given in Section IV-A.
A. Experimental Setup
A 2-DOF serial-chain manipulator was created for the exper-imental tests (see following Fig. 2). The sole purpose of the firstunactuated DOF is to record the orientation and movement ofthe end-effector reference frame, which is fixed to the secondlink. The second DOF is actuated with a Bimba Position Feed-back Cylinder (model #PFC-092-XBL). This cylinder containsan internal LMP for measuring the relative motion, or stroke ofthe rod with respect to the cylinder base.
The first angle is measured using a precision rotary poten-tiometer (BOURNS model 6639S-1–103). The second joint
FOLK AND WOLBRECHT: ACCELEROMETER CONFIGURATION FOR REFERENCE-FRAME-INDEPENDENT LINEAR-STATE ESTIMATION 1133
Fig. 2. Picture of 2-DOF experimental apparatus (top) and close-up view ofmounted accelerometers. Two accelerometers are mounted on the cylinder body,one is mounted on the cylinder rod, and forth is mounted at the end of the secondlink (fixed to the end-effector reference frame).
angle is calculated directly from cylinder stroke length mea-sured by the internal LMP.
The experimental apparatus includes 4 three-axis analog de-vices accelerometers (model #ADXL325EB): two mounted onthe cylinder body, one mounted on the cylinder rod, and onemounted at the end of the second link (fixed to the end-effectorreference frame). For the three accelerometers mounted on thecylinder body and rod, only the acceleration measurementsalong the motion of the rod were used in the 3ACC EKF. Forthe single accelerometer mounted on the end effector, two axisof accelerometer were used for the kinematically dependent EEEKF (the two axes in the plane of motion). The EE EKF is aplanar version of [15], and requires kinematic knowledge of thedevice so the gravity vector may be subtracted out and acceler-ation direction may be determined, a requirement not necessarywith the presented approach (3ACC EKF).
For additional comparison, a linear-velocity sensor (LVS)(TRANS-TEK 100 model #0012–0001) was mounted externallyto the pneumatic cylinder. This sensor directly measures velocityof the cylinder rod with respect to the cylinder base. In addition,this velocity measurement was added to the presented 3ACCEKF in order to create the 3ACC plus velocity sensor EKF(3ACC + VEL EKF), which measures all states locally andfuses them into kinematic agreement.
Data acquisition and control was performed using xPC Tar-get and a multifunction data acquisition card (MeasurementComputing model PCI-DAS1200) with a sampling frequencyof 1000 Hz, a common sampling frequency for robotic andother control applications.
B. Performance of 3ACC EKF With and Without KinematicKnowledge of the Accelerometer’s Local Reference Frame
In the first experiment, the 2-DOF manipulator was manuallymoved through a range of joint angles. The position estimates
Fig. 3. Position estimates from five different state-estimation techniques. De-tailed series descriptions are given in Table I. A zoomed in section of the left-sideplot is given in the right-side plot as indicated by the boxed area and arrow.
during this movement from the different position-estimationmethods are given in the following Fig. 3.
The results from aforementioned Fig. 3 show that the3ACC EKF and EE EKF have comparable performance whenkinematic knowledge of the accelerometers is known. Bothof these methods closely follow the directly measured LMPsignal, and both methods have similar smoothness to the filteredLMP signal, but without the phase lag introduced by the fil-ter. In comparison, the EE EKF without kinematic knowledge(EE EKF without kinematics in Fig. 3) is unable to accuratelyestimate position because the orientation and position of theaccelerometers at the end effector is unknown. Conversely, thiskinematic information is not required for the presented 3ACCEKF method, which measures local relative acceleration, andthus, the performance is not adversely affected.
The velocity estimate results given in the following Fig. 4are similar to those from the position estimate results, with re-spect to state-estimator performance. Both the 3ACC EKF andEE EKF have comparable performance, in both smoothness andphase lag, when kinematic knowledge of the accelerometers isknown. However, the EKF without kinematic knowledge (EEEKF without kinematics) is unable to accurately estimate veloc-ity because the orientation and position of the accelerometers atthe end effector are unknown. Again, this kinematic informationis not required for the presented 3ACC EKF method.
The results from both position estimate (see Fig. 3) and ve-locity estimate (see Fig. 4) comparisons demonstrates how the3ACC EKF is not affected by the movement of the local refer-ence frame, and is able to cancel out the external accelerationfrom the manipulator motion and the acceleration due to gravity.When kinematic knowledge is not available, the EE EKF (EEEKF without kinematics) is not able to distinguished relative
1134 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 6, DECEMBER 2011
Fig. 4. Velocity estimates from five different state-estimation techniques. De-tailed series descriptions are given in Table I. A zoomed in section of the left-sideplot is given in the right-side plot as indicated by the boxed area and arrow.
acceleration from acceleration cause by angular velocity andgravity. The most noticeable inaccuracy is in the velocity estima-tion, which is particularly problematic for velocity-dependentcontrol systems, such as pneumatic systems, where accurate(low noise and phase lag) velocity signals are critical for forceand position control.
C. Performance of 3ACC EKF Versus Direct Velocity Measure-ment Methods
Another, alternate approach is to add a velocity sensor to thesystem and either use direct velocity measurement by itself orinclude it as part of an EKF. Following Fig. 5 shows how both ofthese methods compare to the presented 3ACC EKF and otherstate-estimation techniques.
The results in Fig. 5 illustrate that the velocity estimate fromthe 3ACC EKF is more differentially accurate than measuringvelocity directly, i.e., the direct velocity measurement does notagree with taking the discrete derivative from the directly mea-sured position from the LMP. The 3ACC + VEL EKF correctsthis problem by fusing the local position, velocity, and accel-eration measurements. However, it is clear from the results inFig. 5 that the resulting velocity estimate from the 3ACC +VEL EKF is at best only marginally better than the velocityestimate from the 3ACC EKF. In addition, the velocity sensoris consistently of lower magnitude than the discrete derivativeof the position signal, and thus, may actually be detrimental tostate estimation in some instances. Therefore, it is difficult tojustify the additional expenses, including cost, complexity, andreliability, caused by the addition of a LVS.
Fig. 5. Comparison of four velocity-estimation techniques. Detailed seriesdescriptions are given in Table I. A zoomed in section of the left-side plot isgiven in the right-side plot as indicated by the boxed area and arrow.
V. DISCUSSION AND CONCLUSION
This paper has presented a novel three single-axis accelerom-eter configuration for directly measuring the linear accelerationof a point with respect to a moving reference frame and an EKFthat combines this local relative acceleration measurement witha local position measurement for state estimation. Experimentaltesting has shown several advantages of the presented method.First, the presented method performs as well as standard end-effector-based methods, but without requiring kinematic knowl-edge of the local reference frame. Therefore, this method maybe used in mobile robotic and other applications, where precisecontrol of an actuator is needed, but kinematic knowledge of thelocal reference frame is difficult to accurately obtain.
Measuring relative acceleration in the local reference framehas a second important advantage. Because the approach doesnot require kinematic knowledge of the entire manipulator, stateestimation, and actuator controller development may be devel-oped prior to use in a multi-DOF manipulator. With pneumaticcylinder actuators, for example, force and position control maybe implemented at the cylinder level, allowing the potential de-velopment of a pneumatic cylinder equipped with its own forceand position controller. This integrated pneumatic actuator andcontroller could then be used in applications, where the capabil-ities of pneumatic actuators are desired, but the complexity ofpneumatic control development prohibitively discourages theiruse.
A third advantage to the presented method is that the EKFfuses the data from various sensors into differential agreement,which is assumed in most control stability proofs. In contrast,the experimental data showed that the direct measurement ofvelocity from the LVS was not accurate when compared to thederivative of the LMP position measurement. In addition, the
FOLK AND WOLBRECHT: ACCELEROMETER CONFIGURATION FOR REFERENCE-FRAME-INDEPENDENT LINEAR-STATE ESTIMATION 1135
experimental results also showed that adding the velocity mea-surement into the presented EKF did not significantly improveperformance. Considering the additional cost of the sensor andthe potential decrease in system reliability due to added com-plexity, it is clear that the addition of a LVS is not justified.
A final advantage of the presented method is the direct mea-surement of local relative acceleration. This signal may be usedin the actuator local control law. For example, in pneumaticsystems, acceleration is typically part of the controller feed-back. However, while the end-effector-based EKF can provideposition and velocity measurements to joint and actuator space,acceleration measured at the end effector is difficult to map tojoint or actuator space, and thus is not typically available at theactuator level for control feedback.
As a potential disadvantage, the presented method has theexpense of additional acceleration sensors and A/D conversionchannels. In actual cost, the A/D channels are more significantthan the additional sensors. This can be minimized by subtract-ing the accelerations as presented by analog means. Further-more, more electronics (sensors, etc.) expose the system to ad-ditional potential points of failure. It is believed that these pointsof failure could be minimized by developing a fully integratedsystem. Further testing is needed to evaluate the robustness ofthe proposed system to sensor and analog circuitry.
This paper was motivated by the need for high-quality statesignals (especially velocity) for pneumatic cylinder force andposition control. However, as presented the method of apply-ing three accelerometers in the presented configuration could beapplied to independent LMPs, linear encoders, and other variouslinear-displacement sensors. Such application could produce alinear-displacement sensor capable of producing accurate, low-noise position, velocity, and acceleration signals regardless ofthe intended application and with disregard to potential un-known motions and orientations.
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Brandon G. Folk received the B.S. degree in me-chanical engineering from the University of Idaho,Moscow, where he is currently working towardthe Ph.D. degree in the Mechanical EngineeringDepartment.
His current research interests include robotics,nonlinear and adaptive control, pneumatic control,and combined human–robotic systems.
Eric T. Wolbrecht (M’06) received the B.S. degreein mechanical engineering from Valparaiso Univer-sity, Valparaiso, IN, in 1996, the M.S. degree in me-chanical engineering from Oregon State University,Corvallis, in 1998, and the Ph.D. degree in mechani-cal and aerospace engineering from the University ofCalifornia, Irvine, in 2007.
He is currently an Assistant Professor in theDepartment of Mechanical Engineering, Universityof Idaho, Moscow. His research interests includerobotics, nonlinear and adaptive control, compliant
actuation, motor learning, and neurorehabilitation.Dr. Wolbrecht is a member of the American Society of Mechanical
