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Research Papers

Pose Changes From a Different Point of View

[+] Author and Article Information
Gregory S. Chirikjian

Department of Mechanical Engineering,
Johns Hopkins University,
Baltimore, MD 21218
e-mail: gchirik1@jhu.edu

Robert Mahony

College of Engineering and Computer Science,
The Australian National University,
Canberra ACT 2601, Australia
e-mail: robert.mahony@anu.edu.au

Sipu Ruan

Department of Mechanical Engineering,
Johns Hopkins University,
Baltimore, MD 21218
e-mail: ruansp@jhu.edu

Jochen Trumpf

College of Engineering and Computer Science,
The Australian National University,
Canberra 2601, ACT, Australia
e-mail: jochen.trumpf@anu.edu.au

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 22, 2017; final manuscript received December 22, 2017; published online February 27, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 10(2), 021008 (Feb 27, 2018) (12 pages) Paper No: JMR-17-1315; doi: 10.1115/1.4039121 History: Received September 22, 2017; Revised December 22, 2017

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semidirect product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here, we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semidirect product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing pose changes of its hands.

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Figures

Grahic Jump Location
Fig. 1

Demonstrating the three-frame scenario with a separate observer frame O for a humanoid robot

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Fig. 2

Relationship between the frames O, A, B, and C

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Fig. 3

Translation vectors  OOtA, OOtB, OOtC, AOtB, BOtC

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Fig. 4

Demonstration of the difference between  AWdB and  AWtB, BWdC and  BWtC, AWdC and  AWtC (note that  AWdB=BWdC=AWdC=0)

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Fig. 5

Demonstration of  AOdB, BOdC, AOdC and  AOtB, BOtC, AOtC

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Fig. 6

Transformation of the observer frame from O to 1

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Fig. 7

Coordinates of all vectors computed in frame 1 instead of frame O

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Fig. 8

Conjugation resulting from changing body-fixed frames

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Fig. 9

 AOtA¯+A¯OtB¯=AOtB+BOtB¯

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Fig. 10

Comparison of geodesic trajectories for SE(3) and PCG(3)

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Fig. 11

Comparisons of interpolation using multiple frames and two adjacent frames. Dashed curves: SE(3); Solid curves: PCG(3); Dotted curves: lines that connect poses. (a) Multiple frames and (b) adjacent frames.

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Fig. 12

Comparisons of interpolation by shifting according to the mean of frames. Dashed curves: SE(3); Solid curves: PCG(3); Dotted curves: lines that connect poses. (a) Mean (μ) of frames in SE(3)  (0.96590.2586−0.00857.5−0.25860.9638−0.064615−0.00850.06460.997815) and (b) Mean (μ) of frames in PCG(3) (0.96590.2586−0.00859.6656−0.25860.9638−0.064615.9882−0.00850.06460.997812.2066).

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Fig. 13

Comparisons of interpolation based on different parameterizations of time steps. Dashed curves: SE(3); Solid curves: PCG(3); Dotted curves: lines that connect poses. (a) Fitted curves based on evenly distributed time steps and (b) fitted curves based on distance-related time steps.

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