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Research Papers

# Effective Tool-Point Acceleration of Serial Chain Mechanisms Based on Basic Geometric Transformations

[+] Author and Article Information
Oziel Rios1

Department of Mechanical Engineering, Robotics Research Group, University of Texas at Austin, Austin, TX 78758jrios90610@yahoo.com

Delbert Tesar

Department of Mechanical Engineering, Robotics Research Group, University of Texas at Austin, Austin, TX 78758tesar@mail.utexas.edu

The translational tool-point acceleration is given by $ẍT=JTφ̈+φ̇THTφ̇$ where $HT∊R3×N×N$ is the translational component of the Hessian array or second-order kinematic influence coefficient (17), but recall that the manipulator starts from rest meaning that $φ̇=0$.

The gravity vector is typically $ĝ=(0,0,−9.8)T m/s2$.

Note that in Eq. 20, the sum starts at $j=i$ to include the weight of link $i$. For the actuators, $Laici=0$ for all $i$ since we are assuming that actuator $i$ is located on axis $i$.

In other words, the translational kinematic influence coefficient associated with the center-of-mass of the actuator is zero since its associated position function is zero.

This is equivalent to transforming the kinetic energy into the variables defining the motion of the tool-point and solving for the effective mass term.

For this range of reduction ratios, it is plausible to vary the reduction ratio of the gear train while maintaining its weight and rotary inertia constant (22).

If the force capability is improved by means of increasing the transmission ratio, for example, then the acceleration capability can decrease due to the increase in effective mass (see trend in Fig. 7).

1

Corresponding author.

J. Mechanisms Robotics 1(4), 041004 (Sep 17, 2009) (9 pages) doi:10.1115/1.3204254 History: Received September 22, 2008; Revised April 08, 2009; Published September 17, 2009

## Abstract

In this paper, a method to manage the actuator parameters of a serial chain mechanism composed of revolute joints to achieve improved responsiveness characteristics (acceleration capability) based on the basic geometric parameters of the mechanism is presented. Here, an analytic framework presented by the authors in an earlier work, which exploits the geometric structure of this type of mechanism is extended to address the tool-point mass and acceleration. The manipulator’s geometry is reduced to a set of lengths, which are representative of the mechanical gains associated with the manipulator and they, along with the transmission ratio of the actuators, are used to map the actuator parameters to their effective values at the tool-point where a direct comparison to the task requirements can be made. With this method, minimal computations are required to evaluate the system’s performance since only the forward kinematic computations are required. The effects of the actuator transmission ratio parameter on the effective tool-point force, mass, and acceleration are investigated for a six-DOF serial chain manipulator. Through this case study, it is demonstrated how the transmission ratio is managed to balance the system’s effective tool-point force and mass to obtain an optimal tool-point acceleration. In addition to the investigation of the effects of the actuator parameters, the method is shown to be useful in the solution of the configuration management or modular design problem since the exponential design space can be searched for a globally optimal solution with minimal computations. The goal of the configuration management problem is to quickly configure and/or reconfigure a robotic manipulator from a finite set of actuator modules.

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## Figures

Figure 1

Geometric parameters associated with joint axis i and tool-point

Figure 2

Geometric parameters associated with joint axis i and center-of-gravity of link j

Figure 3

Figure 7

Trends of the (a) effective tool-point force f3eff (N), (b) effective tool-point mass m3eff (kg), and (c) effective tool-point acceleration a3eff(m s−2) of joint axis 3 for varying transmission reduction ratio values G¯3

Figure 4

Effective mass and inertia of an actuator and link pair rotating about the z-axis

Figure 5

Description of the distance Lijc between axis i and jth CM and the angle θijc between the z-axis of frame attached to axis i and the z-axis of frame attached to jth CM

Figure 6

Six-DOF manipulator geometry

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