Research Papers

Improving Techniques in Statically Equivalent Serial Chain Modeling for Center of Mass Estimation

[+] Author and Article Information
Bingjue Li

Department of Mechanical and
Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: lib002@udayton.edu

Andrew P. Murray, David H. Myszka

Department of Mechanical and
Aerospace Engineering,
University of Dayton,
Dayton, OH 45469

Manuscript received September 26, 2014; final manuscript received November 23, 2014; published online December 31, 2014. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 7(1), 011013 (Feb 01, 2015) (10 pages) Paper No: JMR-14-1270; doi: 10.1115/1.4029294 History: Received September 26, 2014; Revised November 23, 2014; Online December 31, 2014

Any articulated system of rigid bodies defines a statically equivalent serial chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. An SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed, followed by the experimental validation of SESC modeling. Second, the problem of generating an SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.

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Fig. 1

A branched-chain system composed of four moving articulated rigid bodies

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Fig. 2

The kinematic and static parameters of the system shown in Fig. 1

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Fig. 3

The SESC model of the branched chain system shown in Fig. 2

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Fig. 4

A two-body system that includes a redundancy in the B matrix when CoM data are only gathered in the X–Y plane

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Fig. 5

An experimental model of an articulated branched-chain system. A moving reference frame is attached at each joint, and the joint labels indicate the axes of rotation and the numbering scheme used in establishing the SESC.

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Fig. 6

(a) The experimental model was posed in a configuration and balanced on a peg supporting the bottom plate and (b) the cylindrical peg used to support the experimental model

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Fig. 7

A top view of the bottom plate of the model. The contact area between the bottom plate and the peg causes deviation of the balancing point; therefore, the average of the maximum and the minimum coordinates of the CoM reading was used as the CoM projection data for each configuration. The location of the model's bottom plate in the measuring plane and the location of the supporting peg in the fixed reference frame.

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Fig. 8

(a) A vertical line is drawn through the CoM projection in the X–Y plane. (b) The model in the same configuration is balanced with a peg supporting one of the links. Another vertical line passing through the contact point is drawn. The intersection of the two lines is the CoM of the model, thus the CoM location along the Z axis is found.

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Fig. 9

A four-body system with the first body fixed on the ground

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Fig. 10

(a) A numerical model of a spatial 11-body-17DOF humanoid-like branched chain developed in matlab and (b) its ideal SESC model S

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Fig. 11

E versus the quantity of experimental data points collected for the spatial humanoid-like model shown in Fig. 10(a)

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Fig. 12

The 90th percentile values of E versus the quantity of experimental data for the same process as shown in Fig. 11

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Fig. 13

A curve fitting of the 90th percentile values of E as shown in Fig. 12

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Fig. 14

Curve fittings of the 90th percentile values of E for the humanoid-like model corresponding to different data errors. (The point indicated by “•” shows that the estimated 90th percentile error of E is about 4% after 54 data points are collected with EC = ±0.4 and EJ = ±1 deg.)

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Fig. 15

E versus number of data for (a) a planar four-body model and (b) a six-body robot-arm-like model with experimental data of various precision

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Fig. 16

The error of the CoM predicted for 300 random configurations using an SESC determined experimentally from another 54 data points for which EC = ±0.4 and EJ  = ±1 deg




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