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Research Papers

# Isotropic Accelerometer Strapdowns and Related Algorithms for Rigid-Body Pose and Twist Estimation

[+] Author and Article Information
Ting Zou

Post-doc Fellow
Centre for Intelligent Machines,
Department of Mechanical Engineering,
McGill University,
e-mail: ting@cim.mcgill.ca

Jorge Angeles

Fellow ASME
Centre for Intelligent Machines,
Department of Mechanical Engineering,
McGill University,
e-mail: angeles@cim.mcgill.ca

In the cited reference, the authors dub their architectures simplectic, which is a misnomer.

Superscripts $⊥$ and $∥$ are to be read “perp” and “par,” respectively.

In mathematical programming, a “simplex” is a polyhedron in $Rn$ with the minimum number of vertices, i.e., n + 1 [17].

This concept refers to the second moment of the array of vertices of the isotropic polyhedral, a.k.a. geometric moment-of-inertia tensor, with three identical eigenvalues.

The axial vector of a 3 × 3 matrix A is defined as $a≡vect(A)≡(1/2)[a32-a23a13-a31a21-a12]T$, a vector invariant of A [16].

The Buckyball is also known as the 60-vertex truncated icosahedron, whose shape appears in soccer balls.

For any Platonic solid, its dual polyhedron is constructed in such way that each vertex of the latter coincides with the centroid of the corresponding face of the former.

1Corresponding author.

Manuscript received April 7, 2013; final manuscript received August 24, 2014; accepted manuscript posted August 27, 2014; published online September 10, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(11), 111003 (Sep 10, 2014) (13 pages) Paper No: JAM-13-1150; doi: 10.1115/1.4028405 History: Received April 07, 2013; Revised August 24, 2014; Accepted August 27, 2014

## Abstract

A novel design of accelerometer strapdown, intended for the estimation of the rigid-body acceleration and velocity fields, is proposed here. The authors introduce the concept of isotropic-polyhedral layout of simplicial biaxial accelerometers (SBA), in which one SBA is rigidly attached at the centroid of each face of the polyhedron. By virtue of both the geometric isotropy of the layout and the structural planar isotropy of the SBA, the point tangential relative acceleration is decoupled from its centripetal counterpart, which is filtered out, along with the angular velocity. The outcome is that the rigid-body angular acceleration can be estimated independent of the angular velocity, thereby overcoming a hurdle that mars the estimation process in current accelerometer strapdowns. An estimation algorithm, based on the extended Kalman filter, is included. Simulation results show an excellent performance of the proposed strapdowns in estimating the acceleration and velocity fields of a moving object along with its pose.

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## References

Yazdi, N., Ayazi, F., and Najafi, K., 1998, “Micromachined Inertial Sensors,” Proc. IEEE, 86(8), pp. 1640–1658.
Barbour, N., and Schmidt, G., 2001, “Inertial Sensor Technology Trends,” IEEE Sens., 1(4), pp. 332–339.
Collin, J., and Lachapelle, G., 2002, “MEMS-IMU for Personal Positioning in a Vehicle—A Gyro-Free Approach,” GPS 2002 Conference (Session C3a), Portland, OR, Sept. 24–27.
Cappa, P., Patanè, F., and Rossi, S., 2008, “Two Calibration Procedures for a Gyroscope-Free Inertial Measurement System Based on a Double-Pendulum Apparatus,” Meas. Sci. Technol., 19(5), pp. 32–38.
Dinapoli, L. D., 1965, “The Measurement of Angular Velocities Without the Use of Gyros,” MS thesis, The Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA.
Pickel, W. C., 2005, “Estimation of Postlaunch Angular Motion for Kinetic Energy Projectiles,” AIAA J. Guid. Control Dyn., 28(4), pp. 604–610.
Pamadi, K. B., Ohlmeyer, E. J., and Pepitone, T. R., 2004, “Assessment of a GPS Guided Spinning Projecting Using an Accelerometer-Only IMU,” AIAA Paper No. 2004-4881.
Tan, C. W., and Park, S., 2005, “Design of Accelerometer-Based Inertial Navigation Systems,” IEEE Trans. Instrum. Meas., 54(6), pp. 2520–2530.
Chen, J. H., Lee, S. C., and DeBra, D. B., 1994, “Gyroscope Free Strapdown Inertial Measurement Unit by Six Linear Accelerometers,” AIAA J. Guid. Control Dyn., 17(2), pp. 286–290.
Tan, C. W., Park, S., Mostov, K., and Varaiya, P., 2001, “Design of Gyroscope-Free Navigation Systems,” IEEE Transportation Systems Conference, Oakland, CA, Aug. 25–29, pp. 286–291.
Wang, Q., Ding, M. L., and Zhao, P., 2003, “A New Scheme of Non-Gyro Inertial Measurement Unit for Estimating Angular Velocity,” 29th Annual Conference of the IEEE Industrial Electronics Society (IECON-2003), Roanoke, VA, Nov. 2–6, pp. 1564–1567.
Lin, P. C., and Ho, C. W., 2009, “Design and Implementation of a 9-Axis Inertial Measurement Unit,” IEEE International Conference on Robotics and Automation, Kobe, Japan, May 12–17, pp. 736–741.
Padgaonkar, A. J., Krieger, K. W., and King, A. I., 1975, “Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers,” ASME J. Appl. Mech., 42(3), pp. 552–556.
Cardou, P., and Angeles, J., 2008, “Angular Velocity Estimation From the Angular Acceleration Matrix,” ASME J. Appl. Mech., 75(2), p. 021003.
Cardou, P., and Angeles, J., 2007, “Simplectic Architectures for True Multi-Axial Accelerometers: A Novel Application of Parallel Robots,” IEEE International Conference on Robotics and Automation, Rome, Italy, Apr. 10–14, pp. 181–186.
Angeles, J., 2007, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 3rd ed., Springer, New York.
Kreyszig, E., 1997, Advanced Engineering Mathematics, Wiley, New York.
Hervé, J. M., 1999, “The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design,” Mech. Mach. Theory, 34(5), pp. 719–730.
Zou, T., and Angeles, J., 2012, “Structural and Instrumentation Design of a MEMS Biaxial Accelerometer,” Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, Montreal, QC, Canada, Technical Report No. TR-CIM-06-12.
Angeles, J., 2010, “On the Nature of the Cartesian Stiffness Matrix,” Ing. Mec. Tecnol. Desarrollo, 3(5), pp. 163–170, available at: http://revistasomim.net/revistas/3_5/art1.pdf
White, A. J., and Young, J. B., 1993, “Time-Marching Method for the Prediction of Two-Dimensional, Unsteady Flows of Condensing Steam,” J. Propul. Power, 9(4), pp. 579–587.
Jer-Nan, J., and Minh, Q., 2004, Identification and Control of Mechanical Systems, Cambridge University, Cambridge, UK.
Baron, L., and Angeles, J., 2000, “The Direct Kinematics of Parallel Manipulators Under Joint-Sensor Redundancy,” IEEE Trans. Rob. Autom., 16(1), pp. 12–19.
Cardou, P., 2008, “Design of Multiaxial Accelerometers With Simplicial Architectures for Rigid-Body Pose and Twist Estimation,” Ph.D. thesis, McGill University, Montreal, QC, Canada.
Baltes, H., Brand, O., Fedder, G., Hierold, C., Korvink, J., and Tabata, O., 2005, Circuit and System Integration, Vol. 2, Wiley-VCH, Weinheim, Germany.

## Figures

Fig. 1

A rigid body carrying n uniaxial accelerometers

Fig. 2

Acceleration field of a rigid body: (a) tangential and (b) centripetal components

Fig. 3

Layout of simplicial accelerometers: (a) SUA; (b) SBA; and (c) STA

Fig. 4

Compliant realization of the Π-joint with: (a) a pair of long beams and (b) four notched hinges

Fig. 5

SBA design with its dimensions

Fig. 6

Two instances of accelerometer strapdowns: (a) nonisotropic and (b) isotropic

Fig. 7

A rigid rotating disk with strapdowns: (a) tetrahedral and (b) general

Fig. 8

Estimated acceleration of the disk of Fig. 7(a) using a tetrahedral SBA strapdown

Fig. 9

Estimated twist of the disk of Fig. 7(a) using a tetrahedral SBA strapdown

Fig. 10

Estimated pose of the disk of Fig. 7(a) using a tetrahedral SBA strapdown

Fig. 11

Estimated acceleration of the disk of Fig. 7(b) using a brick SBA strapdown

Fig. 12

Estimated twist of the disk of Fig. 7(b) using a brick SBA strapdown

Fig. 13

Estimated pose of the disk of Fig. 7(b) using a brick SBA strapdown

Fig. 14

A rigid brick with accelerometer strapdowns: (a) tetrahedral and (b) general

Fig. 15

Measured acceleration of the brick through isotropic SBA strapdown

Fig. 16

Measured twist of the brick through isotropic SBA strapdown

Fig. 17

Measured attitude of the brick through isotropic SBA strapdown

Fig. 18

Measured acceleration of the brick through a brick SBA strapdown

Fig. 19

Measured twist of the brick through a brick SBA strapdown

Fig. 20

Measured attitude of the brick through a brick SBA strapdown

Fig. 21

Illustration of the tetrahedral strapdown with its local and global coordinate frame

Fig. 22

Illustration of the brick strapdown with local and global coordinate frames

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